Set Packing� Partitioning� and Covering

Aspects

Set Packing Partitioning and Covering

Ralf Borndorfer

C L

Aspects

Set Packing Partitioning and Covering

vorgelegt von DiplomWirtschaftsmathematiker

Ralf Borndorfer

Vom Fachb ereich Mathematik der Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

genehmigte Dissertation

Promotionsausschu

Vorsitzender Prof Dr Kurt Kutzler

Berichter Prof Dr Martin Grotschel

Berichter Prof Dr Rob ert Weismantel

Tag der wissenschaftlichen Aussprache Marz

Berlin D

Zusammenfassung

Diese Dissertation b efat sich mit ganzzahligen Programmen mit

Systemen SetPacking Partitioning und CoveringProbleme Die

drei Teile der Dissertation b ehandeln p olyedrische algorithmische und

angewandte Asp ekte derartiger Mo delle

Teil diskutiert p olyedrische Asp ekte Den Auftakt bildet ei

ne Literaturubersicht in Kapitel In Kapitel untersuchen wir

SetPackingRelaxierungen von kombinatorischen Optimierungspro

blemen ub er Azyklische Digraphen und Lineare Ordnungen Schnitte

und Multischnitte Ub erdeckungen von Mengen und ub er Packungen

von Mengen Familien von Ungleichungen fur geeignete SetPacking

Relaxierungen sowie deren zugehorige Separierungsalgorithmen sind

auf diese Probleme ub ertragbar

T eil ist algorithmischen und rechnerischen Asp ekten gewidmet

Wir dokumentieren in Kapitel die wesentlichen Bestandteile ei

nes BranchAndCut Algorithmus zur Losung von SetPartitioning

Problemen Der Algorithmus implementiert einige der theoretischen

Ergebnisse aus Teil Rechenergebnisse fur Standardtestprobleme der

Literatur werden b erichtet

Teil ist angewandt Wir untersuchen die Eignung von Set

PartitioningMetho den zur Optimierung des Berliner Behinderten

fahrdienstes Telebus der mit einer Flotte von Fahrzeugen taglich

etwa Fahrwunsche b edient Der BranchAndCut Algorith

mus aus Teil ist ein Bestandteil eines Systems zur Fahrzeugein

satzplanung das seit dem Juni in Betrieb ist Dieses Sy

stem ermoglichte Verb esserungen im Service und gleichzeitig erhebli

che Kosteneinsparungen

Schlusselb egrie Ganzzahlige Programmierung Polyedrische

Kombinatorik Schnitteb enen BranchAndCut Anrufsammeltaxi

systeme Fahrzeugeinsatzplanung

Mathematics Sub ject Classication MSC C

Abstract

This thesis is ab out integer programs with constraint systems Set

packing partitioning and covering problems The three parts of the

thesis investigate p olyhedral algorithmic and application asp ects of

such mo dels

Part discusses p olyhedral asp ects Chapter is a prelude that sur

veys results on integer programs from the literature In Chapter

we investigate set packing relaxations of combinatorial optimization

problems asso ciated with acyclic digraphs and linear orderings cuts

and multicuts multiple knapsacks set coverings and no de packings

themselves Families of inequalities that are valid for such a relaxation

and the asso ciated separation routines carry over to the problems un

der investigation

Part is devoted to algorithmic and computational asp ects We do cu

ment in Chapter the main features of a branchandcut algorithm for

the solution of set partitioning problems The algorithm implements

some of the results of the theoretical investigations of the preceding

part Computational exp erience for a standard test set from the liter

ature is rep orted

Part deals with an application We consider in Chapter set par

titioning metho ds for the optimization of Berlins Telebus for handi

capp ed p eople that services requests p er day with a eet of

mini busses Our branchandcut algorithm of Part is one mo dule of

a scheduling system that is in use since June and resulted in

improved service and signicant cost savings

Keywords Integer Programming Polyhedral Combinatorics

Cutting Planes BranchandCut Vehicle Scheduling DialARide

Systems

Mathematics Sub ject Classication MSC C

Preface

Asp ects of set packing partitioning and covering is the title of this

thesis and it was chosen delib erately The idea of the thesis is to try

to b end the b owfrom theory via algorithms to a practical application

but the red thread is not always pursued conclusively This resulted in

three parts that corresp ond to the three parts of the b ow and b elong

together but that can also stand for themselves This selfcontainment

is reected in separate indices and reference lists

There is no explanation of notation or basic concepts of optimization

Instead I have tried to resort to standards and in particular to the

book Grotschel Lovasz Schrijver Geometric Algorithms and

Combinatorial Optimization Springer Verlag Berlin

It is p erhaps also useful to explain the system of emphasis that is at

the bottom of the writing Namely emphasized words exhibit either

the topic of the curren t paragraph andor they mark contents of the

various indices or they sometimes just stress a thing

I am grateful to the Senate of Berlins Departments for Science Re

search and Culture and for So cial Aairs that supp orted the Tele

bus pro ject and to Fridolin Klostermeier and Christian Kuttner for

their co op eration in this pro ject Iamindebted to the KonradZuse

Zentrum for its hospitality and for its supp ort in the publication of

this thesis

Iwould like to thank my sup ervisor Martin Grotschel for his example

not only as a mathematician and esp ecially for his patience I also

thank Andreas Schulz and Akiyoshi Shioura who have kindly p ointed

out anumb er of errors in an earlier version of this thesis My friends

and Alexander Martin havehelpedme Norb ert Ascheuer Bob Bixby

with many discussions on asp ects of this thesis and I want to express

my gratitude for this A sp ecial thanks go es to Andreas Lob el for

his friendship and supp ort My last sp ecial thanks go es to my friend

Rob ert Weismantel I simply want to say that without him not only

this thesis would not b e as it is

Ihopethat who ever reads this can prot a little from these notes

and p erhaps even enjoy them

Berlin August Ralf Borndorfer

Contents

Zusammenfassung v

Abstract vii

Preface ix

I Polyhedral Asp ects

Integer Programs

Two Classical Theorems of Konig Intro duction

The Set Packing Partitioning and Covering Problem

Relations to Stable Sets and Indep endence Systems

Blo cking and AntiBlo cking Pairs

Perfect and Ideal Matrices

Minor Characterizations

Balanced Matrices

The Set Packing Polytop e

Facet Dening Graphs

Comp osition Pro cedures

Polyhedral Results on ClawFree Graphs

Quadratic and Semidenite Relaxations

Adjacency

The Set Covering Polytop e

Facet Dening Matrices

Set Packing Relaxations

Intro duction

The Construction

The Acyclic Sub digraph and the Linear Ordering Problem

The Clique Partitioning Multi and Max Cut Problem

The Set Packing Problem

Wheel Inequalities

A New Family of Facets for the Set Packing Polytop e

Chain Inequalities

Some Comp osition Pro cedures

The Set Covering Problem

The Multiple Knapsack Problem

The Programming Problem with Nonnegative Data

Bibliography of Part

Index of Part

xii Contents

II Algorithmic Asp ects

An Algorithm for Set Partitioning

Intro duction

Prepro cessing

Reductions

Data Structures

Probabilistic Analyses

Empty Columns EmptyRows and Row Singletons

Duplicate and Dominated Columns

Duplicate and Dominated Rows

Row Cliques

Parallel Columns

Symmetric Dierences

Column Singletons

Reduced Cost Fixing

Probing

Pivoting

The Prepro cessor

Separation

The Fractional Intersection Graph

Clique Inequalities

Cycle Inequalities

Aggregated Cycle Inequalities

Computational Results

Bibliography of Part

Index of Part

III Application Asp ects

Vehicle Scheduling at Telebus

Intro duction

Telebus

The Vehicle Scheduling Problem

Pieces of Work

Requests

Constraints

Ob jectives

Solution Approach

Transition Network

Decomp osition

Set Partitioning

AVehicle Scheduling Algorithm

Related Literature

Contents xiii

Cluster Generation

Tour Generation

Chaining Network

Tour Enumeration

Heuristics

Computational Results

Clustering

Chaining

Vehicle Scheduling

Persp ectives

Bibliography of Part

Index of Part

Index of Symbols

Curriculum Vitae

xiv Contents

List of Tables

Prepro cessing Airline Crew Scheduling Problems

Estimating Running Times of Prepro cessing Op erations

Analyzing Prepro cessing Rules

Solving Set Partitioning Problems byBranchandCut Default Strategy

Solving Set Partitioning Problems byBranchandCut Separating Aggregated

Cycle Inequalities

Solving Set Partitioning Problems byBranchandBound

Solving Clustering Set Partitioning Problems

Solving Chaining Set Partitioning Problems

Comparing Vehicle Schedules

xvi List of Tables

List of Figures

The KonigEgervary and the Edge Coloring Theorem

Constructing a Column Intersection Graph

A Clique

A Cycle

A Line Graph of a Connected Hyp omatchable Graph

AAntihole

A Wheel

The Antiweb C

C The Web

A ComplementofaWedge

A Chain

The Comp osition of Circulants C

AClaw

The Generalized Antiweb AW

The Generalized Web W

A Cycle of Dipaths

APolyhedron and Its AntiDominant

Constructing a Set Packing Relaxation

AFence

AMobius Ladder of Dicycles

AFence Clique

AMobius Cycle of Dipaths

A Chorded Cycle

Lab eling Lower Triangles

An OddCycle of Lower Triangles

The Oddk Circulant C

The Evenk Circulant C

Lab eling Upp er Triangles

An Odd Cycle of Upp er Triangles

A Wheel

A Wheel and a Cycle of No des and Edges

Two Generalizations of Odd Wheel Inequalities

I A Wheel and a Cycle of Paths of Typ e

A Wheel and a Cycle of Paths of Typ e I I

A Cycle of Cycles

A Chain

The Antiweb C

Applying a Comp osition Pro cedure

Another Comp osition Pro cedure

A Sekiguchi Partitionable Matrix

A Not Sekiguchi Partitionable Matrix and an Aggregated Cycle

xviii List of Figures

The Nonzero Structure of Set Partitioning Problem nw

Storing Sparse Matrices in Column Ma jor Format

Bringing Set Partitioning Problem nw into Staircase Form

Eliminating Column Singletons in the Right Order to Avoid Fill

Clustering Requests in a DialaRide System

ATelebus Picks Up a Customer

Op eration of the Telebus System

Organization of the Telebus System

Increasing Usage and Costs of Telebus

Results of the Telebus Pro ject

Telebus Requests in June

Highways and Ma jor Roads in Berlin

Constructing a Transition Network

Clusters at Telebus

Enumerating Clusters by Depth First Search

Enumerating Tours by Depth First Search

Reducing Internal Travelling Distance by Clustering

FromaTelebus Pro ject Flyer

Part I

Polyhedral Asp ects

Chapter

Integer Programs

Summary This chapter tries to survey some of the main results of the literature for integer

programming problems asso ciated with set packings set partitionings and set coverings

Blo cking and antiblo cking theory the eld of p erfect ideal and balanced matrices and the

results ab out the facial structure of set packing and set covering p olyhedra

Two Classical Theorems of Konig Intro duction

Konigs b o ok Theorie der end lichen und unend lichen Graphen of is the rst systematic

treatment of the mathematical discipline of graph theory Two hundred years after Eulers

famous primer on the bridges of Konigsb erg gave birth to this area of discrete mathematics

it was Konigs aim to establish his sub ject as a branch of combinatorics and abstract set

theory In this spirit he investigated structural prop erties of general and of sp ecial classes of

graphs Among the latter bipartite graphs are the sub ject of two of his most famous theorems

the KonigEgervary and the edge coloring theorem In his own words these results read as

follows

Sachs page of Konig Konig himself made an eort to compile all previous

references on graph theory

Euler attributes the notion of graph theory or geometria situs as he called it to Leibniz and gives some

references in this direction see Euler page of Konig

Konig preface page Second one can p erceive it the theory of graphs abstracting from its

continuosgeometric content as a branch of combinatorics and abstract set theory This book wants to

emphasize this second p oint of view

Translation by the author

Integer Programs

Theorem KonigEgervary Theorem Konig

For any bipartite graph is the minimum number of no des that drain the edges of the graph

equal to the maximum numb er of edges that pairwise do not p ossess a common endp oint

hh ii

Here wesay that the no des A A A drain the edges of a graph if every edge of the

graph ends in one of the p oints A A A

Theorem Edge Coloring Theorem Konig

If at most g edges come together in every no de of a nite bipartite graph G one can sub divide

all edges of the graph into g classes in such away that every two edges that come together

in a no de b elong to dierent classes

theorems can be seen as combinatorial minmax theorems and they are among the Konigs

earliest known results of this typ e Minmax theorems state a duality relation between two

optimization problems one a minimization and the other a maximization problem hence

the name In the KonigEgervary case these optimization problems are the minimum node

covering problem and the maximum matching problem in a bipartite graph the theorem states

that the optimum solutions the minimum covers and the maximum matchings are of equal

size The edge coloring theorem involves also two optimization problems The trivial task

to compute the maximum degree in a bipartite graph is related to the problem to determine

the minimum number of colors in an edge coloring The relation is again that the b est such

values are equal See the top of page for an illustration of the twoKonig theorems

Minmax results are imp ortant from an optimization point of view b ecause they provide

simple certicates of optimality For example to disp erse any doubt whether some given

cover is minimal one can exhibit a matching of the same size The technique works also the

other way round or it can b e used to prove lower or upp er bounds on the size of a minimal

cover or maximum matching resp ectively And most imp ortant of all optimality criteria are

the rst step to design combinatorial optimization algorithms

It go es without saying that the relevance of his theorems was more than clear to Konig

and he devoted two entire sections of his book to their consequences Konig showed for

instance that the p opular but fortunately rarely applied marriage theorem can b e derived

in this way He noticed also that the two theorems themselves are related and proved that

the edge coloring theorem fol lows from the KonigEgervary theorem Reading his b o ok one

has the impression that Konig lo oked at the rst as a weaker result than the latter and

we could nd no evidence that he considered the reverse implication But we know to day

that exactly this is also true It is one of the consequences of Fulkerson s powerful

antiblocking theory develop ed ab out years later that the KonigEgervary and the edge

coloring theorem are equivalent This means that for bipartite graphs not only no de covering

and matching are dual problems as well as edge coloring is dual to degree computation but

going one step further these two minmax relations form again a dual pair of equivalent

erson called it companion theorems as Fulk

See also Konig Theorem XIV page of Konig

See Konig Theorem XI page of Konig

Konig page of Konig Theorem XIV is an imp ortant theorem that can b e applied

to problems of very dierent nature Applications follow Page of Konig Theorem XI

that is equivalent to the edge coloring Theorem XI can b e applied to various combinatorial problems

Applications follow

Konig XI x edge coloring and XIV x KonigEgervary See in b oth cases also the preceeding

paragraphs

Konig page of Konig

Two Classical Theorems of Konig Intro duction

Lets go through an application of antiblo cking theory to the KonigEgervaryedge coloring

setting now to see how this theory works The antiblo cking relation deals with integer

programs of a certain packing typ e and we start by formulating a weighted generalization

of the matching problem in this waythebipartite matching problem BMP Taking A as the

no deedge incidence matrix of the bipartite graph of interest a rowforeach no de a column

for each edge this BMP can b e formulated as the weighted packing problem

BMP max w x Ax x x binary

nonnegative integer Here is a vector of all ones of compatible dimension w is a vector of

weights and taking w is to lo ok for a matching of maximum cardinality The packing

structure in BMP is that the constraint system is and of the form Ax x

Note that matching is a synonym for edge packing hence the name

Now we apply a sequence of transformations to this program Removing the integrality

stipulations taking the dual and requiring the dual variables to b e integral again

T T T T

min y max w x max w x min y

T T T T

y A w Ax Ax y A w

T T T T

y y x x

y integral x integral

we arrive at another integer program on the right This program is the weighted bipartite

node covering problem BCP of edges bynodes

T T T T T T

BCP min y y A w y y integral

BCP is an example of a weighted covering problem which means in general that the con

T T T

straint system is of the form y A w y with a matrix A and arbitrary integer

weights w on the righthand side

But the BCP is for w exactly the no de covering problem of the KonigEgervary theorem

This relation allows us to paraphrase Theorem in integer programming terminology as

follows For w the optimum ob jective values of the packing problem BMP and of the

asso ciated covering problem BCP are equal

The key p oint for all that follows now is that this equality do es not only hold for w but

for any integral vector w In other words a weighted generalization of the KonigEgervary

theorem as ab ove holds and this is equivalent to saying that the constraint system of the

packing program BMP is total ly dual integral TDI This situation a TDI packing system

Ax x with matrix A is the habitat of antiblo cking theory and whenever we can

establish it the antiblo cking machinery automatically gives us a second companion packing

program again with TDI constraint system and asso ciated minmax theorem In the Konig

Egervary case the companion theorem will turn out to be a weighted generalization of the

edge coloring theorem for bipartite graphs

The companion program is constructed as follows We rst set up the incidence matrix B

of all solutions of the packing program ie in our case of all matchings versus edges a row

for each matching a column for each edge This matrix is called the antiblocker of A it

t matrix of the companion packing program and its asso ciated dual serves as the constrain

T T T T

max w x max w x min y min y

T T T T

Bx Bx y B w y B w

T T T T

x y x y

x integral y integral

Integer Programs

The main result of antiblo cking theory is that if the original packing program had a TDI

constraint system the companion packing program has again a TDI constraint system This

means that all inequalities in the sequence hold with equality for all integral weights w

and this is the companion minmax theorem

What do es the companion theorem sayintheKonigEgervary case for w The solutions

of the left integer program in are edge sets that intersect every matching at most once

Sets of edges that emanate from an individual no de have this prop erty and a minutes thought

shows that these are all p ossible solutions w means to lo ok for a largest such set ie

to compute the maximum no de degree this is one half of the edge coloring theorem The

second integer program on the right of provides the second half b ecause it asks for a

minimum cover of edges by matchings But as the matchings are exactly the feasible color

classes for edge colorings the integer program on the right asks for a minimum edge coloring

And arbitrary weights give rise to a weighted generalization of the edge coloring theorem

We can thus saythattheweighted version of the KonigEgervary theorem implies byvirtueof

antiblo cking theory the validity of a companion theorem whichisaweighted generalization

of the edge coloring theorem One can work out that it is p ossible to reverse this reasoning

such that these two theorems form an equivalent pair And one nally obtains the twoKonig

theorems by setting w The reader will have noticed that in contrast to what we have

claimed on page this antiblo cking argument do es not prove the equivalence of the two

unweighted Konig theorems that b oth only follow from their equivalent weighted relatives

Well sometimes its clearer to lie a little

Our discussion of Konigs considerations was already in terms of weighted versions of his

theorems and further generalizations take us directly to to days areas of researchoninteger

programming problems

The rst question that comes up is whether TDI results with dual pairs of minmax theorems

also hold for other matrices than the incidence matrices of bipartite graphs This question

leads to perfect graph theory where Lovasz has shown that dual minmax theorems

on stable sets and clique cov erings on the one hand and cliques and no de colorings on the

other hold exactly for perfect matrices the clique matrices of perfect graphs This famous

result that was conjectured by Berge and is kno wn as the perfect graph theoremdoes

not imply that the four optimization problems that wehave just mentioned can b e solved in

time that is polynomial in the input length of the p erfect graph and the ob jective b ecause

the asso ciated clique matrix and its antiblo cker can b e exp onentially large But exactly this

is nevertheless p ossible Fundamental algorithmic results of Grotschel Lovasz Schrijver

often termed the polynomial time equivalence of separation and optimization and

techniques of semidenite programming were the key innovations for this breakthrough

Another app ealing topic on p erfect graphs and their clique matrices are recognition problems

An imp ortant result in this area which follows from results of Padb erg b but w as

rst stated and proved in a dierent way by Grotschel Lovasz Schrijver is that

the recognition of p erfect graphs is in coNP This question as well as the unsolved problem

whether one can certify in p olynomial time that a given graph is p erfect or weaker whether

a given matrix is p erfect is intimately related to a stronger and also unresolved version

ct graph conjecture states that a graph is p erfect if of Berges conjecture This strong perfe

and only if it do es not contain an odd hole or its complement it is known to hold for several sub classes of p erfect graphs

Two Classical Theorems of Konig Intro duction

Another direction of research considers general matrices that do not corresp ond to clique

matrices of p erfect graphs The LP relaxations of the packing and the dual covering prob

lems asso ciated to such matrices are not integral much less TDI and minmax theorems do

not hold in general To solvesuchpacking problems with LP techniques additional inequal

ities are needed One branch of research pioneered by Padb erg a is concerned with

nding not only any feasible but in a sense best p ossible facet dening inequalities and to

develop computationally useful pro cedures to nd them For sp ecial classes of matrices it

is sometimes not only p ossible to determine some facets but to obtain a complete description

ie a list of al l facet dening inequalities In such cases there is a chance that it is p ossible

to develop p olynomial LP based or combinatorial optimization algorithms for the four opti

mization problems that come up in packing Maximum stable set minimum clique covering

maximum clique and minimum coloring And in very rare instances complete descriptions

giveeven rise to TDI systems with asso ciated minmax theorems

Analogous problems as in the packing case but much less complete results exist for set

covering problems One obtains the four optimization problems of this area by simply reversing

all inequalities in the four packing analogues But this technique do es not carry over to all

theorems and pro ofs It is in particular not true that every covering minmax theorem has an

equivalent companion theorem and the connection to graph theory is much weaker than in

the packing case The well b ehaved matrices are called ideal but there are no algorithmic

results as for p erfect matrices The study of facet dening inequalities for the nonideal case

seems to b e more dicult as well and little is known here but comparable even though more

dicult results exist for the recognition of ideal matrices

Finally one can lo ok at the equality constrained partitioning case that leads to the con

sideration of a certain class of balanced matrices These matrices give rise to partitioning

programs with integer LP relaxations but the balanced matrices are only a sub class of all

matrices with this prop erty A sp ectacular result in this area is the recent solution of the

recognition problem by Conforti Cornuejols Rao There are no investigations to

determine further inequalities for programs with unbalanced matrices b ecause this question

reduces to the packing and covering case

The following eight sections of this chapter give a more detailed survey on results for the set

packing the set partitioning and the set covering problem Section gives basic denitions

and references to survey articles Section describ es the fundamental connections of set

to graph theory and of set covering to indep endence systems Blo cking and anti packing

blo cking theory is visited a rst time in Section This topic extends to Section where

we discuss p erfect and ideal matrices and the asso ciated famous minmax results the p erfect

graph theorem with its manyvariants and the widthlength and max owmin cut prop erties

of ideal matrices Section is ab out the recognition of p erfect and ideal matrices and

closely related their characterization in terms of forbidden minors Balanced matrices are

treated in a separate Section The last two sections survey p olyhedral results Section

deals with the set packing p olytop e and Section with the set covering p olytop e

Integer Programs

The Set Packing Partitioning and Covering Problem

Let A be an m n matrix and w an integer nvector of weights The set packing SSP

the set partitioning SPP and the set covering problem SCP are the integer programs

T T T

SCP min w x SPP min w x SSP max w x

Ax Ax Ax

x x x

n n n

x f g x f g x f g

Asso ciated to these three programs are six p olyhedra

n n

P Aconv fx f g Ax g P Aconv fx R Ax g

n n

P Aconv fx f g Ax g P Aconv fx R Ax g

n n

Q Aconv fx f g Ax g QAconv fx R Ax g

The set packing polytope P A the set partitioning polytope P A and the set covering

polytope Q A are dened as the convex hull of the set of feasible solutions of SSP SPP

and SCP resp ectively the p olyhedra P A P A and QA denote fractional relaxations

fractional set packing polytope etc The fundamental theorem of linear programming that

guarantees the existence of an optimal basic vertex solution allows to state the three integer

programs ab ove as linear programs o ver the resp ectiveinteger p olytop e

T T T

SSP max w x SPP min w x SCP min w x

x P A x P A x Q A

I I

Let us quickly p oint out some technicalities i Empty columns or rows in the constraint

matrix A are either redundant lead to unb oundedness or to infeasibilityandwe can assume

without loss of generality that A do es not contain such columns or rows ii If A do es not

P A and Q A are always nonempty but P A contain empty rows or columns

I I

is p ossible iii By denition P A P A Q A ie it is enough to study P A

I I I

and Q A to know P A iv The set covering p olytop e Q A as we have dened it is

I I

This trick is convenient for duality arguments and b ounded but the relaxation QAisnot

do es not giveaway information b ecause all vertices of QA lie within the unit cub e v The

two packing p olytop es P A and P A are down monotone the covering p olyhedra Q A

I I

and QA are in slightly dierentsenses up monotone vi These observations can b e used

to assume wlog that set packing or covering problems have a nonnegative or p ositive

objective and so for set partitioning problems as well by adding appropriate multiples of rows

to the ob jective vii Similar techniques allow transformations between the three integer

programs see Garnkel Nemhauser and Balas Padb erg for details

All three integer programs have interpretations in terms of hypergraphs that show their

combinatorial signicance and explain their names Namely lo ok at A as the edgeno de

incidence matrix of a hyp ergraph A on the groundset f ng of columns of A with no de

weights w Then the packing problem asks for a maximum weight set of no des that intersects

all edges of A at most once a maximum packing the covering case is ab out a minimum weight

set that intersects each edge at least once a minimum cover or old fashioned transversal

while in the last case a b est partition of the groundset has to b e determined

We distinguish integer programs with variables and integer programs with matrices

Relations to Stable Sets and Indep endence Systems

We suggest the following survey articles on integer programs Fulkerson blo cking

and antiblo cking theory Garnkel Nemhauser Chapter set partitioning set

covering Balas Padb erg applications set packing set partitioning set packing

p olytop e algorithms Padb erg set pac king p olytop e Schrijver blo c king

and antiblo cking p erfection balancedness total unimo dularity extensions Lovasz

p erfect graphs Grotschel Lovasz Schrijver set pac king p olytop e p erfect graphs

Ceria Nobili Sassano set co vering Conforti et al and Conforti Corn uejols

Kap o or Vuskovic p erfect ideal and balanced and matrices Schrijver

Chapter textb o ok and nally Balinski as a historical article

Relations to Stable Sets and Indep endence Systems

We discuss in this section two insights that are the foundations for the combinatorial study

of the set packing and the set covering problem The corresp ondence between set packings

and stable sets that builds the bridge from packing integer programs to graph theory

and the relation of set covering to indep endence systems

x x

x x x

x x

x x x

x x

x f g

x x

Figure Constructing a Column Intersection Graph

We start with set packing Edmonds last two sentences on page came up with

the idea to asso ciate to a set packing problem SSP the following conict or column inter

section graph GA The no des of GA are the column indices of A and there is an edge

between two column no des i and j if they intersect ie A A see Figure The

i j

construction has the prop erty that the incidence vectors of stable sets in GA ie sets of

pairwise nonadjacent no des are exactly the feasible solutions of the packing program SSP

This means that the set packing program SSP is simply an integer programming formulation

of the stable set problem SSP on the asso ciated conict graph GAwithnodeweights w

For this reason we will o ccasionally also denote P Aby P GA

I I

Two consequences of this equivalence are i Two matrices A and A give rise to the

same set packing problem if and only if their intersection graphs coincide ii Every rowofA

is the incidence vector of a clique in GA ie a set of pairwise adjacent no des In particular

GA GA if A is the cliqueno de incidence matrix of al l cliques in GA or of a set

of cliques such that each edge is contained in some clique or of all maximum cliques with

resp ect to set inclusion see Padb erg a Note that the last matrix contains a maximum

of clique information without any redundancies

Set covering is known to b e equivalent to optimization over independence systems see eg

Laurent or Nobili Sassano by the ane transformation y x

Integer Programs

T T T

w ISP max w y min w y

i Ay A I A y

ii y y

iii y y

n n

iv y f g y f g

To see that the program on the right is an optimization problem over an indep endence system

can we have to construct a suitable indep endence system To do this note rst that one

delete from ISP anyrow that strictly contains some other row A matrix without such

redundant rows is called proper Fulkerson Assuming wlog that A is prop er we

can take its rows as the incidence vectors of the circuits of an independence system IA on

the groundset of column indices of A Then the righthand side A I j supp A j

i i

of every constraint i in ISP equals the rank of the circuit supp A and ISP is an integer

programming formulation of the problem to nd an indep endent set of maximum weightwith

resp ect to w in IA

We remark that there is also a graph theoretic formulation of the set covering problem in

terms of a bipartite rowcolumn incidence graph that has been prop osed eg by Sassano

and Cornuejols Sassano

Thinking again ab out the relation of set packing and set covering in terms of stable sets and

indep endence systems one makes the following observations i The stable sets in a graph

form an indep endence system ie set packing is a sp ecial case of set covering with additional

structure ii This argument holds for almost any other combinatorial optimization problem

as well wemention here in particular the generalizedsetpacking problem and the generalized

set covering problem that arise from their standard relatives by allowing for an arbitrary

uniform righthand side see Sekiguchi iii Not every indep endence system can be

obtained from stable sets of some appropriately constructed graph see Nemhauser Trotter

Theorem or P adb erg b Remark for details

Blo cking and AntiBlo cking Pairs

The theory of blocking and antiblocking pairs of matrices and p olyhedra develop ed in Fulk

erson provides a framework for the study of packing and covering problems

that explains why packing and covering theorems o ccur in dual pairs Its technical vehicle is

the duality or polarity who likes the term b etter b etween constraints and verticesextreme

rays of p olyhedra We discuss the basics of the theory here in a general setting for nonnegative

matrices and sp ecialize to the combinatorial case in the following Sections and

The center of the theory is the notion of a blo cking and antiblo cking pair of matrices and

p olyhedra that weintro duce no w Consider a nonnegative not necessarily matrix A and

the asso ciated fractional packing problem FPP and the fractional covering problem FCP

T T

FCP min w x FPP max w x

Ax Ax

x x

Asso ciated to these problems are the fractional packing polytope and the fractional covering

polyhedron that we denote slightly extending our notation by P AandQA resp ectively

Blo cking and AntiBlo cking Pairs

By Weyls description theorem see eg Schrijver Corollary b these b o dies are

generated by their vertices and extreme rays Denote by abl A the matrix that has the

vertices of P Aasitsrows and byblA the matrix that has the vertices of QA as its rows

Then wehave

n T

P Afx R Ax g convvert P A conv abl A

n n T n

QAfx R Ax g convvert QAR convbl A R

Wemust assume that A do es not contain empty columns for the packing equations to hold

abl A is called the antiblocker of the matrix AblA is the blocker of A Asso ciated to these

matrices are again a fractional packing p olytop e and another fractional covering p olyhedron

n T n

abl P Afy R x y x P Ag fy R ablAy g P abl A

T n n

y x x QAg fy R bl Ay g Qbl A bl QAfy R

abl P A is called the antiblocker of the p olytop e P A bl QAistheblocker of QA The

general duality between constraints and verticesextreme rays of p olyhedra translates here

into a duality relation b etween antiblo cking and blo cking matrices and p olyhedra

Theorem Blo cking and AntiBlo cking Pairs Fulkerson

For any nonnegative matrix A holds

T T

iv If a is a vertex of bl QA a x is a i If a is a vertex of abl P A a x is ei

facet of QA In particular ther a facet of P A or can b e obtained

from a facet by setting some lefthand

v bl QAQA side co ecients to zero In particular

ii abl P AP A

vi bl A is prop er and

A A is prop er bl A

iii If A has no empty column so do es abl A

Here abl is short for abl abl and so on Theorem ii and v state that the anti

blo cking relation gives indeed rise to adualantiblocking pair of polyhedra and the blo cking

relation to a dual blocking pair of polyhedra This duality carries over to the asso ciated

matrices Theorem iv and vi establishes a blocking pair of proper matrices The

duality is a bit distorted in the antiblo cking case b ecause the antiblo cking relation pro duces

dominated verticesrows Since only the maximal rows give rise to facets one do es not insist

on including dominated ro ws in a packing matrix and calls two matrices A and B an anti

blocking pair of matrices if the asso ciated packing p olyhedra constitute an antiblo cking pair

Blo cking and antiblo cking pairs of matrices and p olyhedra are characterized by a set of

four relations that provide a link to optimization Let A and B be two nonnegative matrices

and consider the equalities

T T

min y max Bw max y min Bw

T T

y A w

T T

w i mg and so on If holds for al l Here min Bw is short for min fB

nonnegative vectors w wesay that the minmax equality holds for the ordered pair of matrices

A B If holds for al l nonnegative vectors w wesay that the maxmin equality holds for

the ordered pair of matrices A B

Integer Programs

The other two relations are inequalities

T T

max Al max Bw l w min Al min Bw l w

If holds for al l nonnegative vectors w and l we say that the maxmax inequality holds

for the unordered pair of matrices A and B If holds for al l nonnegative vectors w

and l wesay that the minmin inequality holds for the unordered pair of matrices A and B

These equations and inequalities are related to the antiblo cking and the blo cking relation via

appropriate scalings of the vectors w and l such that the ab ove optima become one this is

always p ossible except in the trivial cases w andor l Such a scaling makes w and l

amemb er of the antiblo ckingblo cking p olyhedron These arguments can b e used to prove

Theorem Characterization of Blo cking and AntiBlo cking Pairs Fulkerson

For any pair of nonnegative matrices A and B For any pair of prop er nonnegative matrices A

with no empty columns the following state and B the following statements are equiva

ments are equivalent lent

i A and B are an antiblo cking pair vi A and B are a blo cking pair

ii P A and P B are an antiblo cking pair vii QA and QB are a blo cking pair

iii The minmax equality holds for A B viii The maxmin equality holds for A B

iv The minmax equality holds for B A ix The maxmin equality holds for B A

v The maxmax inequality holds for A and B x The minmin inequality holds for A and B

Theorem b ears on dual minmax results for packing and covering optimization problems

We giveaninterpretation of the antiblo cking part iii and iv of Theorem in terms of

the fractional packing problem the covering case is analogous The minmax equality

can be interpreted as a weighted max fractional packingmin fractional covering theorem

The rows of A are used for covering the rows of B that corresp ond to the feasible solutions of

FPP for pac king If this minmax theorem can b e established antiblo cking theory yields

a second equivalent theorem of the same typ e where the coveringpacking roles of A and B

are exchanged

Perfect and Ideal Matrices

The main p ointofinterest in antiblo cking and blo cking theory is the study of antiblo cking

and blo cking pairs of matrices A and B that are both Saying that a matrix A has a

antiblo cking matrix B is by denition equivalenttointegrality of the fractional packing

p olytop e asso ciated to A a matrix A that gives rise to suchanintegral packing p olytop e

P AP A is called perfect Analogous for covering blo cking matrices corresp ond to

ideal integral covering p olyhedra QAQ A a matrix A with this prop ertyiscalled

By Theorem perfect matrices occur in antiblocking pairs and so do ideal matrices occur

in blocking pairs Asso ciated to an antiblo ckingblo cking pair of p erfectideal matrices is a

pair of equivalent minmaxmaxmin equalities and one can either prove one of the equalities

to establish the second plus the antiblo ckingblo cking prop erty plus p erfectionidealityofa

matrix pair or one can prove one of the latter two prop erties to obtain two minmaxmax min results

Perfect and Ideal Matrices

Antiblo ckingblo cking pairs of p erfectideal matrices often have combinatorial signicance

and this brings up the existence question for combinatorial covering and packing theorems

The minmaxmaxmin equalities and are not of combinatorial typ e b ecause they

allowforfractional solutions of the coveringpacking program But consider stronger integer

forms of these relations for matrices A and B

T T

min Bw max Bw max y min y

T T T T

y A w y A w

T T T T

y y

T m T m

y Z y Z

for al l nonnegative integer vectors w wesay that the strong minmax equality If holds

holds for the ordered pair of matrices A and B this is equivalent to stating that the

packing system Ax x is TDI If holds for al l nonnegative integer vectors w

we say that the strong maxmin equality holds for the ordered pair of matrices A and

B this relation corresp onds to a TDI covering system Ax x The combinatorial

content of these relations is the following The strong minmax equality can be interpreted

as a combinatorial min coveringmax packing theorem for an antiblo cking pair of p erfect

matrices The smallest number of rows of A such that each column j is covered by at least

w rows is equal to the largest packing of columns with resp ect to w where the packings are

enco ded in the rows of B An analogous statement holds in the strong maxmin case for a

blo c king pair of ideal matrices

Wemention two famous examples of such relations to p oint out the signicance of this concept

Dilworths theorem is an example of a wellknown strong minmax equality in the context of

partial ly ordered sets Let A be the incidence matrix of all chains of some given p oset let

B be the incidence matrix of all its antichains and consider the strong minmax equality

for A B It states that for any nonnegative integer vector w of weights asso ciated to the

elements of the p oset the smallest number of chains such that each element is contained in

at least w chains is equal to the maximum w weight of an antichain For w this is

the classical Dilworth theorem and one can generalize it to the weighted case by appropriate

replications of p oset elements the reader mayverify that this is easy The validity of this

weighted generalization of Dilworths theorem implies that A and B form an antiblo cking

pair of p erfect matrices b ecause the strong minmax equality for A B yields trivially the

fractional minmax equality for A B This argument implies in turn the minmax equality for

B A in its fractional form What ab out the strong integer version for B A One can work

out that it holds as wellandthisisnota strikeofluck But lets stop here for the moment

and just consider the combinatorial content of the strong minmax equality for B A This

theorem is identical to the weighted Dilworth theorem except that the words antichain

and chain havechanged their places a combinatorial companion theorem

The most famous example of a strong maxmin equality is probably the max owmin cut

theorem of Ford Jr Fulkerson for t woterminal networks Taking A as the incidence

matrix of all s tpaths versus edges and B as the incidence matrix of all s tcuts versus

y for edges the max owmin cut theorem turns out to b e exactly the strong maxmin equalit

A B Hence the incidence matrices of s tpaths and s tcuts in a twoterminal network

form a blo cking pair of ideal matrices Can one also pro duce a companion theorem byinter

changing the roles of paths and cuts as we did with the antichains and chains in Dilworths

theorem The answer is yes and no One can in this particular case but not in general

Integer Programs

We have already hinted at one of the main insights of antiblo cking theory in the Dilworth

example and we state this result now The p erfection of a matrix A is equivalent to the validity

of the strong minmax equalityforA and abl A which is itself equivalent to the validityofa

companion minmax theorem for abl A and A

Theorem Strong MinMax Equality Fulkerson

Let A b e a matrix without empty columns The following statements are equivalent

v The strong minmax equality holds for i A is p erfect

A abl A

ii abl A is p erfect

iii The system Ax x is integral

vi The strong minmax equality holds for

x is TDI abl A A iv The system Ax

Interpreting this result in terms of the stable set problem see Section weenter the realm

of perfect graph theory A minutes thought shows that the only candidate for a anti

blo cker of the incidence matrix B of all stable sets of some given graph G is the incidence

matrix A of all cliques versus no des Now consider the two p ossible strong minmax equations

the optima of the four asso ciated optimization problems are commonly denoted by

T T

G min y G min y

T T T T

y A w y B w

T T T T

y y

T T

y integral y integral

G max Bw G max Aw

w w

G is called the weighted clique covering number of G G is the weighted stability

number G the weighted coloring numberand Gtheweighted clique number With

w w

this terminology the strong minmax equality for A B translates into the validity of the

equation G Gforany nonnegativeinteger vector w and a graph with this prop erty

is called pluperfect Similarly a pluperfect graph satises the second strong minmax

and plup erfect equality G G forall w Z and a pluperfect graph is b oth

w w

Theorem reads in this language as follows

Theorem Plup erfect Graph Theorem Fulkerson

A graph is plup erfect if and only if it is plup erfect if and only if it is plup erfect

This theorem can also b e stated in terms of complement graphs by noting that plup erfection

G This equivalent of a graph G is equivalent to plup erfection of the complement graph

plup erfect if and only if its complement is version is A graph is

One of the big questions in this context and the original motivation for the development

of the entire antiblo cking theory was the validity of Berge s famous perfect graph

conjecture The conjecture claimed a stronger form of the plup erfect graph theorem where

w is not required to run through all nonnegativeinteger vectors w but only through all

vectors In exactly the same way as in the plup erfect case this concept gives rise to perfect

perfect and perfect graphs hence the conjectures name Fulkersons idea to prove it was

to show its equivalence to the plup erfect graph theorem to establish this it is enough to

prove the following replication lemma Duplicating a vertex of a perfect graph and joining

the obtained twovertices by an edge gives again a p erfect graph The replication lemma and

hence the conjecture was proved by Lovasz and shortly after the result had b ecome

known also byFulkerson

Perfect and Ideal Matrices

Theorem Perfect Graph Theorem Lovasz

p erfect if and only if it is p erfect if and only if it is p erfect if and only if it is A graph is

plup erfect

There is also a complementversion of the p erfect graph theorem Agraphis p erfect if and

only if its complement is And let us further explicitly state an integer programming form

of the p erfect graph theorem that will turn out to have a blo cking analogon We include a

strong version of the maxmax inequality with identical vectors w and l also proved by

Lovasz and F ulkerson

Theorem Perfect Graph Theorem Lovasz Fulkerson

For matrices A and B without empty columns the following statements are equivalent

i A and B are an antiblo cking pair

ii The strong minmax equality holds for A B and all nonnegativeinteger vectors w

iii The strong minmax equality holds for B A and all nonnegativeinteger vectors w

iv The strong minmax equality holds for A B and all vectors w

v The strong minmax equality holds for B A and all vectors w

vi The maxmax inequality holds for A and B and all nonnegativeinteger vectors w and l

vii The maxmax inequality holds for A and B and all vectors w l

Here wehave used the expression the strong minmax equality holds in an obvious sense

slightly extending our terminology A third interesting linear programming form of the p erfect

graph theorem is again due to Lovasz

Theorem Perfect Graph Theorem Lovasz

A matrix A without empty columns is p erfect if and only if the linear program max w x

Ax x has an integer optimum value for all vectors w

Lets take a break from antiblo cking and p erfect graphs at this p oint and turn to the blocking

case Unfortunately the antiblo cking results of this section do not all carry over It is not

true and the main dierence b etween blo cking and antiblo cking theory that the integralityof

the fractional covering p olyhedron corresp onds to a TDI constraint system neither is it true

that the strong maxmin inequalityforA B implies the strong maxmin equalityfor B A see

Fulkerson for a coun terexample And there are also no results that compare to p erfect

graph theory b ecause there is no suitable graph version of the set covering problem

The other Theorems and have analogues that are due to Lehman

pro ofs of these dicult results are given in Padb erg from a p olyhedral p oin t of view

and Seymour from a hyp ergraph p ointof view We state them in the following two

theorems where we adopt the conventions that Theorem iii and that is

an integer Theorem

Theorem WidthLength Prop erty of Ideal Matrices Lehman

For matrices A and B the following statements are equivalent

i A and B are a blo cking pair

ii The minmin inequality holds for all nonnegativeinteger vectors w and l

iii The minmin inequality holds for all vectors w and l restricted to co ecients

and at most one o ccurrence of another co ecient that is equal to the number of

co ecients minus one The fourth typ e of co ecients is solely needed to exclude the

incidence matrices of degenerate pro jective planes see the following Section

Integer Programs

Theorem Max FlowMin Cut Prop erty of Ideal Matrices Lehman

A matrix is ideal if and only if the linear program min w x Ax x has an integer

optimum value for all vectors w

The names for these results come from Lehmans terminology his widthlength inequality is

the same as the minmin inequalitythemax owmin cut equality is the maxmin equality

Generalizing the concepts of p erfection and ideality to matrices we enter an area of

research that is related to the study of total ly unimodular matrices It is b eyond the scop e of

this chapter to discuss these elds or integralTDI systems in general surveys on these

skovic topics are given in Padb erg a and Conforti Cornuejols Kap o or Vu

Minor Characterizations

Both the p erfect graph theorem and the max owmin cut characterization of ideal matrices

have alternativeinterpretations in terms of matrix minors and in the antiblo cking case also

of graph minors that we discuss in this section The study of minors b ears on the recognition

problem for p erfect and ideal matrices

We start in the antiblocking setting Consider the p erfect graph theorem in its linear pro

gramming form and note that setting an ob jective co ecient w to zero has the same

eect on the optimum ob jective value as removing column A from the matrix A Equiva

lentlywe could removenodej from the column intersection graph or yet another equivalent

v ersion we could intersect the fractional packing p olytop e P A with the hyp erplane x

and eliminate co ordinate j The op eration that wehave just describ ed is called a contraction

of co ordinate or column j of the matrix A or of the intersection graph GA or of the frac

tional packing p olytop e P A and the resulting matrix or graph or p olytop e is a contraction

minor of the original ob ject With this terminology considering all ob jectives is the same

as considering ob jective for all contraction minors and one obtains various minor forms of

the p erfect graph theorem by replacing the expression for all vectors w with for all

contraction minors and w For example Theorems and translate in dierent

ways into the following minor results

Theorem Perfect Graph Theorem Lovasz

x A matrix A without empty columns is p erfect if and only if the linear program max

A x x has an integer optimum value for all contraction minors A of A

Theorem Perfect Graph Theorem Lovasz

The following statements are equivalent for a graph G

i G is p erfect iv G G jV G j for all minors G

of G

ii G G for all minors G of G

Here a minor is always a contraction minor iii G G for all minors G of G

The contraction technique can be used also in the blocking scenario to deal with the zero

ob jective co ecients in Theorem A little more dicult is the treatment of the

co ecien ts w amounts to forcing x to one this eect can also b e obtained byremoving

j j

column j from the matrix A as well as all rows that A intersects or by an intersection

of the fractional covering p olyhedron QA with the hyp erplane x and a subsequent

elimination of co ordinate j This op eration is called a deletion of co ordinate or column j of

the matrix A or the p olyhedron QA and its result is a deletion minor It is straightforward

Minor Characterizations

to show that contraction and deletion commute and one can thus call the matrix A the arises

bycontracting and deleting some set of co ordinates of A acontractiondeletion minor of A

This nomenclature gives again rise to a numb er of minor theorems for ideal matrices like

Theorem Minor Characterization of Ideal Matrices Lehman

A matrix is ideal if and only if the linear program min x A x x has an integer

optimum value for all contractiondeletion minors A of A

The minor characterizations for p erfect and ideal matrices b ear on the recognition problems

for these classes Given a matrix A is it p erfectideal or not It is not known whether

any of the recognition problems is in NP or not but Theorems and give a rst

coNP answer Recognizing p erfect and ideal matrices is in coNP if the input length is

assumed to b e O n m ie if we consider A the input Just exhibit a minor such that

or fail and verify this by solving a linear program This result is not very deep however

b ecause one do esnt need the p erfect graph theorem or the max owmin cut characterization

to come up with p olynomial certicates for the existence of a fractional basic solution of an

explicitly given linear system

Anyway researchers are not satised with results of this typ e and we explain now why this

is so for the p erfection test The problem is that the recognition of imperfect matrices do es

not carry over to the recognition of imperfect graphs The reason is that although we could

verify a clique matrix of a graph as imp erfect in p olynomial time this do es not help much

for an eective investigation of some given graph b ecause a clique matrix has in general

already exp onential size in the enco ding length of the graph From this point of view a

coNP complexity result as ab ove seems to b e cheating what we really want are algorithms

with running time p olynomial in the number of vertices columns of A Seymour

And nothing else but exactly this is in fact p ossible One can devise such algorithms for the

verication of imp erfection as well as for the verication of nonideality the latter in a sense

that is yet to b e made precise

The metho ds that resolve these questions are based on the concepts of minimal ly imperfect

or almost perfect and minimal ly nonideal or almost ideal matrices that are not p er

fectideal themselves but any of their deletioncontractiondeletion minors is Obviously

any imp erfectnonideal matrix must contain such a structure and a recognition algorithm

can in principle certify p erfection by making sure that no such minor exists imp erfection

the ideality test One approach to the recognition problem is by exhibiting one and so for

hence to study the structure of minimally imp erfect and nonideal matrices This structure is

still not fully understo o d but to a signicant extent and there are in particular complete

characterizations of minimally imp erfect and minimally nonideal matrices and of p erfect and

ideal matrices in terms of forbidden minors A nal terminological remark As usual there

are also minimal ly imperfect or almost perfect graphs and the same concepts exist for the

fractional packing and covering p olyhedra that are called almost integral

We b egin with results on minimal imperfection where the matrix structures of interest have

the following app earance Wesaythatanm n matrix A has property if

i A contains a regular n n matrix A with column and row sums all equal to

ii each row of A which is not a row of A is either equal to some row of A or has row

sum strictly less than

The matrix A that is obviously unique up to permutations of rows whenever it exists is

called the core of A and denoted by core A

Integer Programs

Theorem Minimally Imp erfect Matrices Padb erg b

An m n matrix A is minimally imp erfect if and only if

i A has prop erty where n mo d and either n or bn c

ii A has no m k contraction minor A with prop erty for any kn and any such

that k

An m n matrix A is p erfect if and only if A do es not contain any m k contraction minor

A having prop erty for k min fm ng and either k or bk c

This theorem makes some progress toward the coNP complexity part of the recognition

problem for p erfect graphs b ecause a core has an enco ding length that is p olynomial in n

and lo oks like a go o d candidate to certify prop erty for some contraction minor A of

the only implicitly known clique matrix A of some given graph G The only problem that

remains is to verify that some matrix A is a core of A In other words How do es one

prove that all cliques in Gsupp A of size are already contained in A and that there are

no larger ones The answer to this question is based on strong structural prop erties of

dual pairs of minimal ly imperfect matriceshow could it b e dierent

To start note that thecoreofa minimally imp erfect matrix A with prop erty pro duces

a fractional vertex x core A of the almost integral p olytop e P A

Padb erg has shown that this is the only fractional vertex And much more is true

Theorem Pairs of Minimally Imp erfect Matrices Padb erg b

Let A be an m n matrix and let B abl A be the integral part of its antiblo cker

Supp ose A is minimally imp erfect with prop erty Then

i B is also minimally imp erfect

ii A has prop erty and B has prop erty where n

n n

A and B have unique cores that satisfy the matrix equation core Acore B E I

iii P A has the unique fractional vertex x

x is adjacent to precisely n vertices of P A namelytherows of core B

Moreover P AfAx x x g

Here E is a matrix of al l ones I is the identity matrix and the matrix equation in ii is

supp osed to b e understo o d mo dulo suitable column and row p ermutations

Theorem has interesting consequences Note that part iii states that all that misses to

make an almost integral packing p olytop e integral is one simple rank facet This situation can

comeupintwoways The rst case is when A is not a clique matrix of its conict graph GA

ie some clique row is missing As A is minimally imp erfect it must have prop erty

n n

GAmust b e a clique and the missing rowis x The second and exciting case is when

A is a clique matrix Then we see from Theorem the following

i G GA has exactly n maximum cliques of size G and exactly n maximum

stable sets of size G the incidence vectors of these maximum cliques and stable

sets are linearly indep endent Each maximum clique intersects all but exactly one

maximum stable set its socalled partner and vice versa

ii For every no de j G j can be partitioned into maximum cliques of size and

maximum stable sets of size where n

Here e denotes the unit vector that has a one in co ordinate j andG j is the minor that

arises from G by contracting no de j i is derived from column j of the matrix equation

core A core B e ii using Theorem iv

Minor Characterizations

A graph that satises the strong condition ii on the preceding page is called partitionable

Note that for such a graph G G and G must hold and since G G

n n partitionable graphs are imp erfect by virtue of Theorem But it is

easy to verify that a graph or a contraction minor of a graph is partitionable and this nally

proves that p erfection of a graph is a prop erty in coNP This complexity result was rst

hel Lovasz Schrijver stated and proved in a dierentway byGrotsc

Theorem Recognition of Perfect Graphs Padb erg b Grotschel

Lovasz Schrijver The recognition problem for p erfect graphs is in coNP

But is that all that one can derive from Padb ergs strong conditions i and ii One can

not help thinking that they stop just by a hair short of a much more explicit characterization

of all minimally imp erfect matrices which is a long standing research ob jective In fact only

two innite but simple classes of minimally imp erfect matrices are known The circulants

C k that are the incidence matrices of odd holes that we denote with the same

symb ol and their antiblo ckers abl C k the incidence matrices of the odd antiholes

the complements of the o dd holes Is that all The strong perfect graph conjecture of Berge

which is p erhaps the most famous op en question in graph theory claims that it is

If so o dd holes and antiholes furnish simple minor certicates of imp erfection But there is

more It do es not seem to be completely out of the question to detect the presence or the

absence of odd holes and antiholes in p olynomial time although nob o dy knows for now if

this is p ossible or not But if the strong p erfect graph conjecture holds and if the recognition

problems for o dd holes and antiholes can b e solved in p olynomial time as well these results

together would solve the recognition problem for p erfect graphs

Ch vatal p ointed out that the strong p erfect graph conjecture holds if one can show

that every minimally imp erfect graph G contains a spanning circulant C ie the

no des of G can be numb ered such that any successive no des i i

indices taken mo dulo form a clique here we denote G G When

Padb ergs conditions b ecame known there was some hop e that they would b e strong enough

to establish this circulant structure in every minimally imp erfect graph But Bland Huang

Trotter showed that one can not prove the strong p erfect graph conjecture in this

way b ecause Padb ergs condition i follows from ii and the partitionable graphs that

satisfy ii do not all contain spanning circulants C

We turn now to the minimal ly nonideal matrices where minor characterizations are known

that are similar to the packing case but more complicated We start with the analogon of

the imp erfection prop erty Wesayanm n matrix A has property if

n n

tains a regular n n matrix A with column and row sums all equal to i A con

ii each row of A which is not a row of A is either equal to some row of A or has row

sum strictly larger than

The matrix A is again unique up to permutations of rows whenever it exists and it is also

called the core of A and denoted by core A

Unlike in the packing case there is however an innite class of minimally nonideal matrices

that do not have constant row and column sums These incidence matrices of degenerate

projective planes p oints versus lines read

where I denotes the n n identity matrix

Integer Programs

Theorem Minimally Nonideal Matrices Lehman Padb erg

If a prop er m n matrix is minimally nonideal then either A J or

i A has prop erty where n mo d

ii A has no m k contractiondeletion minor A with prop erty for any k n and

any suchthat k mo d

An m n matrix A is ideal if and only if A do es not contain any m k contractiondeletion

minor A having prop erty for k min fm n g

The requirement that A is prop er can also be removed but then we must change i from

A J into A contains J and some additional redundantrows Note also that wehave

n n

not claimed an equivalence as for prop erty

As the minimally imp erfect matrices o ccur in antiblo cking pairs so do their minimally non

ideal relatives

Theorem Pairs of Minimally Nonideal Matrices Lehman see also

Padb erg and Seymour

Let A be a prop er m n matrix and let B bl A be the integral part of its blo cker

Supp ose A is minimally nonideal Then

i B is also minimally nonideal

ii Either

a A B J

b QA has the unique fractional vertex x n n n n

x is adjacent to precisely n vertices of QA namely the rows of B

Moreover Q AfAx x n x x n g

I j

c A has prop erty and B has prop erty where n r r min f g

A and B have unique cores that satisfy the matrix equation core Acore B E rI

d QA has the unique fractional vertex x

x is adjacent to precisely n vertices of QA namely the rows of core B

Moreover Q AfAx x x g

The assumption that A is prop er can again be removed as in Theorem Compare also

the co ecients in the lefthand side of the additional facet in Theorem ii b to the

ob jective co ecients in Theorem iii to see that the fourth typ e of ob jective co ecients

the n was only needed to deal with the degenerate pro jective planes J

Seymour used Lehmans minor c haracterization for which he also gives a pro of

to establish that ideality is a coNP prop erty in a sense that can b e seen as the analogon of

Theorem on the recognition of p erfection Seymour views the m n matrix A of

interest as the incidence matrix of a hyp ergraph that should have an enco ding length that

is p olynomial in the number n of elements This creates the problem that the enco ding length

of an m n matrix A is in general certainly not p olynomial in n Seymour assumes thus

that A is given in the form of a lter oracle that decides in constant time whether a given

vector contains a row of A or not Calling this oracle a number of times that is p olynomial

in n one can certify the existence of blo cking matrices core A and core B with prop erties as

in Lehmans Theorem that ensure that A is nonideal

Balanced Matrices

Theorem Recognition of Ideal Matrices Seymour

The recognition problem for ideal matrices that are given by a lter oracle is in coNP

There are some further results toward a more explicit characterization of minimally nonideal

matrices Lehman gave three innite families of minimally nonideal matrices The

incidence matrices of the degenerate pro jective planes J which are selfdual in the sense

they coincide with their blo ckers ie J bl J the odd circulants C k and

n I n

their blo ckers bl C k that coincide via p ermutation of rows and columns with the

circulants C k k But dierent to the packing case many more minimally nonideal

matrices are known

First researchers have compiled a substantial but nite list of exception matrices that do

not b elong to the three innite classes of Lehman The incidence matrix of the Fano plane

B C

is one such exception matrix see Cornuejols Novik and Lutolf Margot for

comprehensive lists But the situation is more complicated than this b ecause further innite

classes of minimally nonideal matrices have been constructed For example Cornuejols

Novik pro veandgive a reference to a similar result that one can add a row e e e

i j

ij ktoany o dd circulant C k k where j i and k j are b oth o dd

and doing so one obtains a minimally nonideal matrix

Do es all of this mean that the set of minimally nonideal matrices is just a chaotic tohuwab ohu

Cornuejols Novik say no and argue that all minimally nonideal matrices in the

known innite nonLehman classes have core C k or C k k This means

geometrically that the asso ciated fractional covering p olyhedra arise from QC k or

QC k k by adjoining additional integral vertices Or to put it dierently the

crucial part of a minimally nonideal matrix is its core and there if one forgets ab out the

exception list only the three Lehman classes have b een encountered These ndings motivate

the following conjecture that can b e seen as the covering analogon of the strong p erfect graph

conjecture

Conjecture Ideal Matrix Conjecture Cornuejols Novik

There is some natural number n such that every m n minimally nonideal matrix A with

n n has core either C k or C k k

Balanced Matrices

Perfect and ideal matrices were dened in terms of integral p olyhedra their characterization

through forbidden minors was and still is a ma jor research problem The study of balanced

matricesthatwere invented by Berge go es the other way round This class is dened

in terms of forbidden minors and one investigates the combinatorial and p olyhedral conse

quences of this construction It turns out that these prop erties subsume all characteristics of

perfect and ideal matrices and balanced matrices give in particular rise to a multitude of

combinatorial packing and covering problems But not only do results from p erfect and ideal

matrix theory carry over There are additional genuine consequences of balancedness that

Integer Programs

include TDIness of b oth asso ciated covering systems and b ear on combinatorial partitioning

theorems And there is another recent sp ectacular result that do es have no parallel yet

Balanced matrices can b e recognized in p olynomial time

It is not the aim of this section to giveanoverview on the entire eld of balanced matrices

to say nothing of their generalizations andor the connections to total ly unimodular

matrices Suchsurveys can b e found in Padb erg a Conforti et al and Conforti

Cornuejols Kap o or Vuskovic we summarize here just some basic results

A matrix A is balanced if it do es not contain an o dd square minor with row and column sums

all equal to two or equivalentlya row and column p ermutation of the circulant C k

In the context of balancedness it is understo o d that minors are not restricted to contraction

and deletion minors instead any subset I of rows and J of columns of A induces a minor

A A As an immediate consequence every such minor A of a balanced matrix A must

also be balanced so is the transp ose A and so is also any matrix that arises from A by

replicating one or several columns any number of times Note that the excluded odd hole

minors C k are one half of the known structures that cause imp erfection and note

also that a balanced matrix do es not only contain no o dd hole contraction minor but no o dd

hole at all ie no o dd hole as any minor The p ossible existence of such dierent but similar

forbidden minor characterizations for balanced and p erfect matrices allows to view the study

of balancedness as a precursor to a p ossible future branch of p erfect matrix and graph theory

after a successful resolution of the strong p erfect graph conjecture

Back to the present and actually years to the past it is easy to see that the edge

no de incidence matrices of bipartite or equivalently colorable graphs are balanced and

balancedness is in fact a generalization of the concept of colorability to hypergraphs The

connection b etween matrices and colorings of hyp ergraphs arises from an interpretation

of the rst as incidence matrices of the latter that go es as follows We asso ciate to a

yp ergraph H H A that has the rows of A as its no des and the columns as matrix A the h

edges H is called balanced if and only if A is Hyp ergraphs can b e colored just like graphs

A node coloring of H assigns a color to each no de such that no edge contains only no des of

a single color the chromatic number H isthe minimum numb er of colors in such a no de

coloring and H is colorable if H It is not so obvious that colorability leads again

back to balancedness but exactly this was Berge s idea and his motivationtointro duce

the whole concept

Theorem Balancedness and Colorability Berge

A matrix A is balanced if and only if H A is colorable for all minors A of A

Many combinatorial prop erties of bipartite graphs carry over to their balanced hyp ergraph

relatives These similarities arise from or are reected in who likes this b etter analogous

symmetries b etween the total ly unimodular and balanced incidence matrices of bipartite and

balanced hyp ergraphs that are stressed in the minor presentation of the following theorem

Theorem Balanced Matrices Berge Fulkerson Homan Opp en

heim For a matrix A the following statements are equivalent

i A is balanced iv P A is integral for all minors A of A

ii A is p erfect for all minors A of A v QA is integral for all minors A of A

iii A is ideal for all minors A of A vi P A is integral for all minors A of A

This is just custom cf the Konig examples of Section the transposed waywould b e feasible as well

Balanced Matrices

We do not delve further into the relations between total unimo dularity and balancedness

here and consider instead the amazing connections to p erfection and ideality The balanced

matrices are exactly those that have only p erfect or ideal minors This has two consequences

First balanced matrices inherit the prop erties of their p erfect and ideal sup erclasses for every

minor which includes in particular all combinatorial minmax and maxmin results Second

Theorem can b e extended bymany other equivalent characterizations of balanced ma

trices in combinatorial p olyhedral and in integer programming terminology just like in the

theory of p erfect and ideal matrices We give three examples to illustrate these p oints

The rst example is another combinatorial minmax characterization of balancedness that we

derive with p erfect matrix techniques Consider the strong minmax equality for A abl A

any minor of A Interpreting this relation in terms of and ob jective w where A is

the hyp ergraph H A is to say that for any partial subhyp ergraph H A of H A the

maximum size of a matching edge packing is equal to the minimum size of a transversal

the equivalence of this relation with balancedness is Berge s Theorem

Example two is an alternative integer programming characterization of balanced matrices

that we obtain from transformations of Theorem ii Namely this statement is equiva

lenttosaying that the integer program max x A x x hasaninteger optimum value

for any minor A of Awhich holds if and only if the linear program max b x Ax w x

n m

has an integer optimum value for any b f g and w f g This is true if and

T T T

only if the dual program min y w y A b y has an integer optimum value for any

n m

b f g and w f g here is not considered to b e an integer and this holds if

T T T T T

program min y y A b y w has an integer optimum value for and only if the

n m

any b f g and w f g The equivalence of this last statement with balancedness is

Berge s Theorem

As a third and last example we show that balanced hyp ergraphs have the Hel ly property

T T

The transp ose A of a balanced matrix A is also balanced hence A is p erfect hence it is a

clique matrix of a graph but the cliques of a graph have the Helly prop ertythat if any two

of a set of cliques have a common vertex they all have a common vertex and the same holds

for the edges of a balanced hyp ergraph this is Berge s Prop osition

We turn next to two prop erties of balanced matrices that are genuine in the sense that

they do not have this inheritance avour TDIness of balanced covering and their blo cking

systems and a strengthening of this last result to one of the rare and precious combinatorial

partitioning maxmin theorems

Theorem TDI Balanced Covering and Blo cking Systems Fulkerson Ho

man Opp enheim

If A is a balanced matrix the strong maxmin equality holds for A bl A and for bl A A

Hence the balanced matrices satisfy an integrality relation that do es not hold in the general

blo cking case Toavoid misunderstandings we p oint out that the blo cker of a balanced matrix

ulkerson Homan Opp enheim for a coun terexample is in general not balanced see F

It is surprising and remarkable that the strong maxmin equalityforblA A can in a certain

sense b e strengthened further into a combinatorial max partioningmin covering theorem

Theorem Partition into Transversals Berge

T T

Let A b e a balanced matrix and B blA its blo cker Then

T T

max y min A

T T T T T

y B y y integral

Integer Programs

To state Theorem in terms of hyp ergraphs note that the blo cker B of A is the incidence

matrix of all transversals of H Aversus no des eachrow is a transversal each column a no de

Then the theorem states the following If k min A is the minimum size of an edge of a

balanced hyp ergraph H A there exist k transversals in H A that partition the vertices or

to put it dierently there is a k coloring of H A suchthateachedgecontains a no de of each

color this is Berge s Theorem

We have seen by now that balanced matrices have analogous but stronger combinatorial

prop erties than p erfect and ideal ones and this trend continues in the study of the recognition

problem The scenario diers slightly from the one for p erfection and ideality testing though

First we explicitly know the complete innite list of all forbidden minors Second there

is no controversy ab out using the matrix itself as the input to the recognition algorithm

Nob o dy has suggested a graphical or other representation of an m n balanced matrix that is

p olynomial in n and mn is accepted as just ne an enco ding length In this setting one of the

most startling results on balanced matrices was the recent construction of an algorithm that

recognizes this class in p olynomial time by Conforti Cornuejols Rao This algorithm

is based on decomp osition metho ds that recursively break a matrix into elementary

pieces in such a way that the balancedness of the whole is equivalent to balancedness of

the pieces and such that the pieces are of combinatorial t yp es whose balancedness can be

established or disproved The recognition of the pieces is based on earlier work on classes of

socalled totally balanced strongly balanced and linearly balanced matrices

Theorem Recognition of Balancedness Conforti Cornuejols Rao

The recognition problem for balanced matrices is in P

Like for p erfect and ideal matrices there is a new branch of research that investigates the

more general class of balanced matrices Conforti Cornuejols sho w for instance

that the memb ers of this class can also b e characterized in terms of colorability and that the

asso ciated packing covering and partitioning system are TDI even in arbitrary mixes An

overview on balanced matrices can b e found in the survey article Conforti Cornuejols

Kap o or Vuskovic

We close this section with a remark on the integralityoffractional set partitioning polytopes

By Theorem vi the balanced matrices form a class that giv es rise to integral p olytop es

of this typ e like p erfect and ideal matrices do to o but these are not all matrices with this

prop erty For a trivial example consider the matrix

C B

that is comp osed from an imp erfect top and a nonideal b ottom It is easy to see that A is

neither p erfect nor ideal nor balanced but one can verify that the fractional set partitioning

p olytop e P A is integral P A consists in fact of the single p oint e We see that the

o ccurrence of forced variables allows to blow up a matrix with all kinds of garbage and

diculties of this sort are the reason why there is no minor theory for matrices with integer set partitioning p olytop es

The Set Packing Polytop e

The set packing problems of the previous sections were almost always assumed to have a

constraint matrix that is p erfect nowwe turn to the general case with arbitrary matrices

Such matrices lead to nonintegral systems Ax x that do not suce to describ e the

set packing p olytop e P A The polyhedral study of general set packing p olytop es aims at

identifying the missing inequalities and at developing metho ds for their eective computation

Such knowledge of the facial structure of set packing p olytop es is useful in three ways To

develop p olynomial time algorithms for classes of stable set problems to derive combinatorial

minmax results and computationally in branchandcut co des for the solution of set packing

or set partitioning problems Letussayaword ab out each of these p oints

The link between polynomial time algorithms and facial investigations is a fundamental al

gorithmic result of Grotschel Lovasz Schrijver that is often termed the polynomial

time equivalence of separation and optimization It is based on the concept of a separation

x as input and decides if it is con oracle for a p olyhedron P that takes an arbitrary p oint

tained in P or if not returns an inequality that separates x and P The theory asserts that

whenever such an oracle is at hand one can optimize over P in oracle polynomial time where

each call of the oracle is counted as taking constant time When the separation can also be

done in p olynomial time this results in a p olynomial optimization algorithm even and in

particular when a complete description of P by linear inequalities has exp onential size And

it turns out that one can construct such p olynomial separation oracles for the set packing

p olytop es of quite some classes of graphs most notably for p erfect graphs

Combinatorial minmax results require explicit complete descriptions by TDI systems It is

theoretically easy to make a linear system TDI but it is dicult to obtain systems of this

typ e with combinatorial meaning In fact b esides p erfect and line graphs there seems to

b e only one class of o dd K free graphs where a combinatorial minmax result is known

of set packing inequalities go es to the other extreme Anything go es The computational use

valid inequalities can b e used as well as facet dening ones and whether exact separation is

always preferable to heuristics well its wiser not to enter this discussion

We try to survey in this section the main results of the p olyhedral approach to the set

packing problem The organization of the section is as follows Subsection intro duces

the concept of facet dening graphs and gives a list of known such structures as well as

of graphs where these inequalities yield complete descriptions Subsection deals with

comp osition pro cedures that construct from simple inequalities more complicated ones Some

results on a sp ecial class of claw free graphs are collected in Subsection Quadratic and

semidenite approaches are treated in Subsection The nal Subsection states

some adjacency results that b ear on primal algorithms

Some basic prop erties of set packing p olytop es for reference in subsequent subsections are

Observation Dimension Down Monotonicity Nonnegativity

Let A b e a matrix and P A b e the asso ciated set packing p olytop e

i P A is full dimensional

ii P A is down monotone ie x P A y P A for all y x

I I I

In particular all nontrivial facets of P A have all nonnegative co ecients

iii The nonnegativity constraints x induce facets of P A

j I

The theory works also for convex b o dies

Integer Programs

Facet Dening Graphs

There are three general techniques to nd valid or facet dening inequalities for the set packing

p olytop e The study of facet dening graphs the study of semidenite relaxations of the set

packing p olytop e and the study of combinatorial relaxations We discuss in this section the

rst technique semidenite relaxations are treated in Subsection and combinatorial

relaxations in Chapter and in particular in Section

The p olyhedral study of general set packing p olytop es through classications of graphs initi

ated byPadb erg a is based on the down monotonicityofP A Namely this prop erty

implies that if H GW is some no de induced subgraph of some given graph G V E

T V

and the inequality a x is valid for P G fx R j x j W g and has a

I j j

for all j W it is also valid for P G The consequence is that substructures of a graph

giverisetovalid inequalities for the set packing p olytop e of the whole graph a relation that

can b e stressed byidentifying the p olytop es P G fx R j x j W g and P H

I j I

and wewant to use this notation here and elsewhere in this section

An investigation of the rules that govern the transfer of inequalities from set packing sub

and vice versa leads to the concepts of facet dening graphs and p olytop es to the whole

lifting see Padb erg a We say that a no de induced subgraph H GW of G denes

the facet a x if this inequality is essential for P H Now among all no de induced

subgraphs H GW of G are those of particular interest that are minimal in the sense that

they give rise to a facet for the rst time This is not always the case If H GW denes

the facet a x for P GW it is p ossible that there is a smaller subgraph G U GW

U W suchthat a x denes already a facet of P GU If this is not the case for all

U W such that jU j jW j the subgraph GW is elementary plusminus one no de

and said to produce a x see Trotter and if this prop erty extends to any subset

U W the subgraph GW is said to strongly produce the inequality Having mentioned

these concepts we do however restrict our attention in the sequel to facet dening graphs

and refer the reader to the survey article of Padb erg for a discussion of facet pro ducing

graphs Moving in the other direction again from small to large the question of what kind

of extensions of valid inequalitiesfacets from subgraphs result in valid inequalitiesfacets for

set packing p olytop es of sup ergraphs is precisely the lifting problem that we discuss in the

next section

We give next a list of facet dening classes of graphs For each such class L one can try to

determine a corresp onding class of Lperfect graphs whose asso ciated set packing p olytop es

can be describ ed completely in terms of L plus the edge inequalities where appropriate

This concept invented byGrotschel Lovasz Schrijver provides a general technique

to identify classes of graphs with p olynomially solvable stable set problems Namely to

establish such a result one merely has to prove that the inequalities from L can b e separated

in p olynomial time Our list includes also these results as far as we are awareofthem

Edge Inequalities Asso ciated to eachedgeij of a graph G V E is the edge inequality

x x Edge inequalities are sp ecial cases of clique inequalities and inherit the face

i j

tial prop erties of this larger class see next paragraph The edge perfect graphs are exactly

the bipartite graphs without isolated no des and these have p olynomially solvable stable set

the system of edge inequalities plus the nonnegativity in problems For general graphs G

equalities AGx x denes an edge relaxation of P A This relaxation has b een

investigated by a number of authors including Padb erg a and Nemhauser Trotter

and displays some initially promising lo oking prop erties Namely P AG has only

The Set Packing Polytop e

half integral vertices all comp onents are or only and stronger all integer com

p onents of a solution of the asso ciated fractional set packing problem have the same value

in some optimal integral solution and can thus be xed Unfortunately this almost never

happ ens in computational practice and neither do es it happ en in theory Pulleyblank

proved that the probability that the edge relaxation of a set packing problem with w on

a random graph has an all optimal solution tends to one when the numb er of no des tends

to innity And this is not only asymptotically true For n the probability of a single

integer comp onent is already less than

Padb erg a A clique in a graph G Clique Inequalities Fulkerson

V E is a set Q of mutually adjacent no des see Figure Asso ciated to such a structure

is the clique inequality

Figure A Clique

The supp ort graphs of Clique inequalities are trivially facet dening Moreover Fulkerson

and Padb erg a have shown that such a constraint induces also a facet for the

stable set p olytop e of a sup ergraph if and only if the clique is maximal with resp ect to set

inclusion in this sup ergraph By denition the clique perfect graphs coincide with the p erfect

graphs Separation of clique inequalities is NP hard see Garey Johnson but this

complexity result is irrelevant b ecause Grotschel Lovasz Schrijver have shown that

the clique inequalities are contained in a larger class of p olynomially separable orthogonality

inequalitiesthatwe will discuss in Subsection This implies that the stable set problem

for p erfect graphs can b e solv ed in p olynomial time This result one of the most sp ectacular

advances in combinatorial optimization subsumes a myriad of statements of this typ e for

sub classes of p erfect graphs see Grotschel Lovasz Schrijver for a surv ey

Odd Cycle Inequalities Padb erg a An odd cycle C in a graph G V E

consists of an o dd number k of no des k and the edges i i for i k

where indices are taken mo dulo k see Figure Any additional edge ij between two

no des of a cycle that is not of the form i i is a chord An o dd cycle without chords is

Asso ciated to a an odd hole the o dd holes coincide with the circulant graphs C k

not necessarily chordless o dd cycle C on k no des is the odd cycle inequality

x jC j

Figure A Cycle

Integer Programs

Padb erg asho wed that the supp ort graph of an o dd cycle inequalityis facet dening if

and only if the cycle is a hole note that the only if part follows from minimal imp erfection

Grotschel Lovasz Schrijver Lemma gave a p olynomial time algorithm to

separate o dd cycle inequalities such that the stable set problem for cycle plus edge perfect

graphs that are also called tperfect t stands for trou the Frenchword for hole is solvable

in p olynomial time Series paral lel graphs graphs that do not contain a sub division of K as

a minor are one prominent class of cycle plus edge p erfect graphs This was conjectured by

Chvatal who showed that the stable set problem for w can b e solved in p olynomial

time and proved by Boulala Uhry a short pro of was given by Mahjoub In fact even

more is true and for a larger class Gerards proved that the system of nonnegativity

edge and odd cycle inequalities is TDI for graphs that do not contain an odd K ie a

sub division of K such that each face cycle is odd This gives rise to a min cycle and edge

coveringmax no de packing theorem Perfect graphs line graphs see next paragraph and

Gerardss class seem to b e the only instances where sucha minmax result is known Alist

of further cycle p erfect graphs can b e found in Grotschel Lovasz Schrijver

Taking the union of clique and o dd cycle inequalities one obtains the class of hperfect graphs

see again Grotschel Lovasz Schrijver for more information

Blossom Inequalities Edmonds The matchings in a graph H V E are in

onetoone corresp ondence to the stable sets in the line graph LH E fij j k E g

of H Asso ciated to such a linegraph LH istheblossom inequality

C C

x jV j

Figure A Line Graph of a Connected Hyp omatchable Graph

Edmonds Pulleyblank sho wed that a blossom inequalityis facet dening for P LH

if and only if H is connected and hypomatchable If we denote by H the maximum size

of a matching in a graph H this graph is hyp omatchable if H H i holds for all

contractions H i of the graph H It is known that a graph H V E is connected

composition E C where C and hyp omatchable if and only if it has an open ear de

is an o dd hole and each C is a path with an even number of nodes v v and distinct

i k k

i i

V C fv v gseeLovasz Plummer endno des v v suchthat V C

j i

i i

Theorem and Figure Separation of blossom inequalities is equivalent to a minimum

odd cut problem see Grotschel Lovasz Schrijver page for which Padb erg

Rao gave a p olynomial time algorithm Edmonds has shown that the stable

set p olytop e of a line graph is completely describ ed by the nonnegativity blossom and the

x for all i V this means that the class of blossom and clique inequalities

e i

clique perfect graphs subsumes the class of line graphs These arguments yield a p olynomial

time algorithm for the stable set problem in line graphs the matching problem in graphs

alternative to the celebrated combinatorial pro cedure of Edmonds Finallywe mention

The Set Packing Polytop e

that Cunningham Marsh ha ve shown that the ab ovementioned complete description

of the set packing p olytop e of a line graph is even TDI which results in a combinatorial min

packingmax covering theorem for edgesblossoms and cliques in graphs

Odd Antihole Inequalities Nemhauser Trotter An odd antihole C is the

complement of an odd hole see Figure the odd antiholes coincide with the circulants

C k k Asso ciated to an o dd antihole on k no des is the odd antihole inequality

C i

Figure AAntihole

Odd antihole inequalities are facet dening by minimal imp erfection As far as we know

no combinatorial separation algorithm for these constraints is known but the odd antihole

inequalities are contained in a larger class of matrix inequalities with N index that can

be separated in p olynomial time see Lovasz Schrijver we will discuss the matrix

inequalities in Subsection These results imply that stable set problems for antihole

perfect graphs can b e solved in p olynomial time

Wheel Inequalities A wheel in a graph G V E is an o dd cycle C plus an additional

no de k that is connected to all no des of the cycle see Figure C is the rim of the

wheel no de k is the hub and the edges connecting the no de k and i i k

are called spokes For such a conguration wehavethewheel inequality

kx x k

Figure A Wheel

Note that wheel inequalities can have co ecients of arbitrary magnitude

A wheel inequality can be obtained by a sequential lifting see next subsection of the hub

into the odd cycle inequalityfor the rim Trying all p ossible hubs this yields a p olynomial

time separation algorithm for wheel inequalities An alternative pro cedure that reduces

wheel separation to odd cycle separation can be found in Grotschel Lovasz Schrijver

Theorem Hence the stable set problem for wheel perfect graphs is solvable in

p olynomial time

Generalizations of wheel inequalities that can b e obtained by sub dividing the edges of a wheel

w ere studied by Barahona Mahjoub who derive a class of K inequalities see the

corresp onding paragraph in this subsection and by Cheng Cunningham who give

also a p olynomial time separation algorithm for two such classes

Integer Programs

We nally refer the reader to Subsection of this thesis where we show that simple as

well as generalized wheels b elong to a larger class of o dd cycles of paths of a combinatorial

rank relaxation of the set packing p olytop e the inequalities of this sup erclass can b e separated

in p olynomial time We remark that the wheel detection pro cedure of Grotschel Lovasz

Schrijver Theorem is with this terminology exactly a routine to detect cycles of

paths of length with one hub endno de

Antiweb and Web Inequalities Trotter Antiweb is a synonym for circulant

see Figure and a web is the complementofanantiweb see Figure Obviouslyevery

odd hole is an antiweb and every o dd antihole is a web An o dd antihole is also an antiweb

but the classes of antiwebs and webs do in general not coincide in fact Trotter pro ved

that an antiweb is a web if and only if it is a hole or an antihole The inequalities asso ciated

to C n k andC n k C n k are the antiweb inequality and the web inequality

x k

x bnk c

iC nk

Figure The Antiweb C

Figure The Web C

An antiweb C n k isfacet dening if and only if either k n or k and n are relatively prime

a web C n k denes a facet if and only if either k or k and n are relatively prime As

far as weknown no p olynomial time separation algorithm for these classes themselves or any

sup erclass is known

Wedge and K Inequalities Giles Trotter Barahona Mahjoub

To construct a wedge one pro ceeds as follows Take a wheel K sub divide its spokes not

the rim and at least one sub division must really add a no de suchthateach face cycle is o dd

and take the complement the resulting graph is a wedge see Figure for a complement

the set of no des E that have an even of a wedge If we sub divide the no des of a wedge into

distance from the original rim no des of the wheel and the set of remaining no des O the

wedge inequality states that

E f g

X X

O f g

x x

i i

iE iO

Figure A ComplementofaWedge

The Set Packing Polytop e

The wedges are facet dening see Giles Trotter Nothing seems to b e known ab out

the separation of this class

By construction the complements of wedges as in this thesis are sp ecial sub divisions of K

All sub divisions of K have b een analyzed by Barahona Mahjoub It turns out that

complete descriptions of the set packing p olytop es asso ciated to arbitrary sub divisions of K

can b e obtained by means of classes of K inequalities The separation of K inequalities

do es not seem to have b een investigated

Chain Inequalities Tesch Ak chain H is similar to the antiweb C k

the dierence is that the twochords k and k are replaced with the single edge

k see Figure This structure gives rise to an inequality for the set packing

p olytop e The chain inequality states that

Figure A Chain

A k chain is facet dening if and only if k mo d Nothing is known ab out the

separation problem

Comp osition of Circulant Inequalities Giles Trotter A composition of

circulants is constructed in the following way Cho ose a p ositiveinteger k letn k k

set up the inner circulant C C n k and the outer circulant C C n k with

no de sets V f n g and V f n g and add all edges ii ii k

for all no des i V indices taken mo dulo n The graph that one obtains from an application

of this pro cedure for any p ositive k is a comp osition of circulants that is denoted by C see

Figure Asso ciated to such a structure is the composition of circulants inequality

X X

k x k x k k

i V

Figure The Comp osition of Circulants C

It is known that comp osition of circulant inequalities are facet dening

Integer Programs

Further Inequalities We close our list of facet dening inequalities for the set packing

p olytop e with some p ointers to further classes

An enumeration of all facets of the set packing p olytop es asso ciated to certain claw free

graphs see next subsection of up to no des has b een done bybyEulerLeVerge

We nally refer the reader to Section of this thesis where we present a class of facet dening

cycle of cycles inequality as an example of a metho d to derive facet dening inequalities from

a rank relaxation of the set packing p olytop e

Wehave seen that most of the facet dening graphs of our list app eared in pairs of graph and

complement graph that give b oth rise to facets and one could thus be lead to b elieve that

some yet to be made precise principle of this sort holds in general Padb erg oers

some sucient conditions in this direction but also p oints out that graphs lik e the line graph

in Figure have complements that are not facet pro ducing no facet dening inequalityof

the asso ciated set packing p olytop e has full supp ort

Our discussion of facet dening graphs would not b e complete without mentioning the neces

sary and sucient conditions that havebeenderived for structures of this typ e It is hard to

come up with interesting characterizations of general constraints and the literature fo cusses

on the already notoriously dicult class of rank inequalities or canonical inequalities as they

are also called Denoting us usual the stabilitynumber or rank of a graph G by G the

rank inequality that is asso ciated to G is

x G

A necessary condition for a rank and more general for any inequality to dene a facet is

Observation Connectedness of a Facets Supp ort

Let G b e a graph and P G the asso ciated set packing p olytop e If a x denes a facet

of P Gitssupp ort graph Gsupp a is no deconnected

Observation which is a sp ecial case of the more general Theorem to b e discussed

in the next section is as far as we know the only general condition that is known the

criteria that follow apply to the rank case with all one co ecients

We start with a sucient condition of Chvatal His criterion for facetial rank inequal

ities is based on the concept of critical edges in a graph G V E Namely an edge ij E

is called critical if its removal increases Gs rank ie if G ij G A graph G

itself is called critical if all of its edges are critical

Theorem Rank Inequalities from Critical Graphs Chvatal

Let G V E b e a graph and E b e the set of its critical edges If the graph G V E

is connected the rank inequality x G is facet dening

The reader can verify that most of the rank inequalities in this sections list satisfy the criterion

of Theorem in fact most have even critical supp ort graphs but this condition is not

necessary see Balas Zemel for a coun terexample

A set of further conditions suggested by Balas Zemel makes use of the notion of a

G critical cutset in a graph G V E ie a cut W such such that G W

In words A cutset is critical if its removal increases the rank

The Set Packing Polytop e

Theorem Critical Cutsets Balas Zemel

Let G be a graph and P G be the asso ciated set packing p olytop e If the rank inequality

for G denes a facet of P Gevery cutset in G is critical

Balas Zemel give an example that shows that this condition is not sucient But

it is p ossible to obtain a complete characterization of those rank facets that arise from facet

dening subgraphs of a graph

Theorem Extension of Rank Facets Balas Zemel

Let G b e a graph P G b e the asso ciated set packing p olytop e W V some subset of no des

of G and let the rank inequality x GW b e facet dening for P GW Then

i I

The rank inequality x GW denes a facet for P G if and only if the cutset

i I

j with resp ect to the graph GW fj g is not critical for every j W

It has been p ointed out by Laurent that Theorems and carry over

to the more general context of rank facets of set c overing polytopes see also Section For

the notion of critical cutsets this corresp ondence is as follows If we interpret the stable

sets in a graph G as the indep endent sets of an indep endence system see Subsection

Theorem says that V is nonseparable while stating that all cutsets j with resp ect to

the graphs GW fj g are not critical as in Theorem is equivalentto W being closed

Comp osition Pro cedures

In the preceding Subsection we have studied and accumulated a list of facet dening

graphsthathave a lo cal relevance in the sense that they are facet dening for their asso ciated

set packing p olytop es In general the given graph will rarely be of one of the sp ecial facet

dening classes but it is not only p ossible but as we known from the minor investigations

of Section inevitable that a given graph contains imp erfect substructures of such typ es

Then by down monotonicity the asso ciated inequalities carry over from the set packing

p olytop es of the subgraphs to the whole

The pro cedure that we have just describ ed is a simple example of a constructive approach

is the following Given validfacet to the study of the set packing p olytop e The idea here

dening inequalities for one or several small graphs compose validfacet dening inequali

ties for a bigger graph In this waywe can build on analytic classications of facet dening

graphs and synthesize global inequalities from elementary pieces

In this subsection we survey two composition procedures of this typ e The lifting metho d and

the study of the p olyhedral consequences of graph theoretic operations

Sequential Lifting Padb erg a The sequential lifting methodthatwas intro duced

by Padb erg a in connection with odd cycle inequalities applied to arbitrary facets of

set packing and set covering p olyhedra by Nemhauser Trotter and further extended

to arbitrary p olytop es by Zemel provides a to ol to build iteratively facets for the

set packing p olytop e P G asso ciated to some graph G from facets of subp olytop es of the

form P GW

Theorem Sequential Lifting Padb erg a Nemhauser Trotter

Let G V E be a graph and P G the asso ciated set packing polytope Let further

W fw w g V be some subset of no des of G that is numbered in some arbitrary

order let W V n W b e the complementof W and let a x b e a facet dening inequality

for P G W I

Integer Programs

Determine numbers R for i k by means of the recursion

max a x x i k

i j j

xP GW fig

Then

The inequality a x x is facet dening for P G

i i I

The ordering of the no des in W is called a lifting sequence is a lifting problem the

is a lifting of the numbers are the lifting coecients the inequality a x x

i i i

inequality a x and the whole pro cedure is referred to as lifting the variables or no des

in W into the inequality a x

Some simple properties of the lifting pro cess are the following If we start with a nonnegative

inequality whichwe assume in the sequel all lifting co ecients will b e nonnegativeaswell

and the righthand side of the original inequalityisan upper bound on the value of eachof

them Taking a lower bound for the value of some lifting co ecient is called a heuristic lifting

step if we do that one or several times the resulting inequality will in general not be facet

dening but it will b e valid Next note that dierentchoices of the lifting sequence give rise

to dierent liftings that have however an identical core a x We remark in this context

that one can also consider the p ossibility to compute several or all lifting co ecients at once

an idea that is called simultaneous lifting see again Zemel

We have already encountered a prominent example of a lifting A wheel inequality can be

obtained by lifting the hub into the o dd cycle inequality that corresp onds to the rim

Sequential lifting is a p owerful conceptual to ol that oers an explanation for the app earance

of facet dening inequalities of general set packing p olytop es Such inequalities frequently

resemble the pure facet dening substructures as in Subsection but with all kinds of

additional protub erances the ab errations can b e understo o d as the results of sequential lift

ings We remark that one do es in general not obtain all facets of a set packing p olytop e P G

from sequential liftings of facets of subp olytop es namely and by denition when the graph G

itself is facet pro ducing examples of facet pro ducing graphs are o dd holes

Turning to the algorithmic side of lifting we note that the lifting problem is again a set

packing problem one for each lifting co ecient So lifting is in principle a dicult task But

the pro cedure is very exible and oers many tuning switches that can be used to reduce

its complexity in rigorous and in heuristic ways First note that when the righthand side

is bounded the lifting problem can be solved by enumeration in pseudo polynomial time

ie time that is p olynomial in the size of the data and the value of For instance clique

inequalities ha ve a righthand side of one and so will be all lifting co ecients sequentially

lifting a clique inequality is simply the pro cess to extend the clique with additional no des in

the order of the lifting sequence until the clique is maximal with resp ect to set inclusion and

this is easy to do in p olynomial time In a similar fashion one can come up with p olynomial

time lifting schemes for antihole inequalities etc all for a xed lifting sequence Second

there are many degrees of freedom for heuristic adjustments One can switch from exact to

heuristic lifting when the lifting problems b ecome hard stop at any point with a result in

not hand make choices in an adaptive and dynamical way etc To put it short Lifting is

the algorithmic panacea of facet generation but it is a useful and exible to ol to enhance

the qualityofany given inequality Some applications of lifting in a branchandcut co de for

set partitioning problems and some further discussion on computational and implementation issues can b e found in Section of this thesis

The Set Packing Polytop e

The last asp ect that we consider here is that the lifting metho d oers also an explanation for

the diculties that one encounters in classifying the facets of the set packing p olytop e It is

extremely easy to use the pro cedure to construct examples of arbitrarily complex inequalities

with involved graphical structures and any sequence of co ecients Do es this mean that

the attempt to understand the facial structure of set packing p olytop es by analysis of small

structures is useless Mayb e but mayb e things are not as bad Padb erg argues

that small facet dening graphs may in a statistical sense give rise to reasonable fractional

relaxations of general set packing p olytop es It is however known that there is no p olynomial

time approximation algorithm for set packing see eg HougardyPromel Steger

Graph Theoretic Op erations We consider in the following paragraphs comp osition pro

cedures that are based on graph theoretic operations Taking one or several graphs p ossibly

of sp ecial typ es w e glue these pieces together to obtain a new graph p ossibly again of a

sp ecial typ e Studying the p olyhedral consequences of such an op eration one tries to de

rive i analogous pro cedures for the comp osition of validfacet dening inequalities or more

ambitious ii complete descriptions for the set packing p olytop e of the comp osition from

complete descriptions for the pieces

Extensions The rst op eration that we consider is the extension of a graph with additional

no des Sequential lifting is an example of this doing when wereverse our p oint of view from

topdown to b ottomup If wedonot lo ok at the seed graph GW of Theorem as

a subgraph of a bigger graph that is given in advance but as a graph of its own the graph

theoretic op eration b ehind each lifting step turns out to be the addition of a single no de

Adding bigger structures results in sp ecial simultaneous lifting pro cedures As an example

we mention a pro cedure of Wolsey and P adb erg who consider the extension of a

graph G with a K A single no de is joined to every no de of G with a path of length Some

asp ects of this pro cedure are discussed in Subsection of this thesis and wemention here

only that Padb erg has shown that the metho d can not only be used to extend facet

dening graphs but to construct facet producing graphs see this sections intro duction that

give rise to facets with arbitrarily complex co ecients

Substitutions This is a second group of powerful graph theoretic manipulations One

takes a graph selects some no de or subgraph substitutes another graph for this comp onent

and joins the substitution to the whole in some way

A rst and imp ortant example of such a pro cedure is due to Chvatal who considered

the replacemen tofanodev of a graph G V E by a second graph G V E node

substitution The graph G that results from this op eration is the union of G v and G

with additional edges that join all no des of G to all neighb ors of v in G Note that no de

substitution subsumes the multiplication or replication of a no de to a clique of Fulkerson

and Lovasz which plays a role in the theory of p erfect graphs Further substituting

for the two no des of an edge yields the sum sometimes also called join graphs G and G

but wewant to use this term later in another way and substituting G for every no de of G

the lexicographic product or composition of G and G No de substitution has the following

p olyhedral consequences

Theorem No de Substitution Chvatal Let G and G b e graphs and let

A x b x and A x b x b e complete descriptions of P G and P G Let

I I

v be a node of G and G b e the graph that results from substituting G for v Then

Integer Programs

The system

X X

a x a a b x b b i j

u u

j i j

ju iv iu

u V

u v

is a complete description of P G

Note that this system is of p olynomial size with resp ect to the enco ding length of the starting

systems A x b x and A x b x

Other authors have considered similar op erations Wolsey obtains facet lifting results

from studying the replacement of a no de with a path of length and of an edge with a path

division dierent from Chvatal s no de substitution these paths of length edge sub

are not connected in a uniform way to the original graph Some discussion of the rst of these

pro cedures can b e found in Subsection of this thesis

Op erations related to paths have also b een considered by Barahona Mahjoub They

transform facets using subdivisions of stars ie simultaneous replacements of all edges that

are incident to some xed no de with paths of length and replacements of paths of length

with inner no des of degree by edges contraction of an odd path the reversal of edge

sub division

Sub divisions of edges and stars are intimately related to the class of K inequalities see

Subsection Namely Barahona Mahjoub haveshown that all nontrivial facets

of P G for such a graph arise from a clique K inequality by rep eated applications of

these op erations The typ es of inequalities that one can pro duce in this way form the class

of K inequalities

Joins The op erations that we term here joins comp ose a new graph from two or more given

graphs in a way that involves an identication of parts of the original graphs Join op erations

often have the app ealing prop erty that they can not only be used for comp osition but also

for decomp osition purp oses because the identication comp onent is left as a ngerprint in

the comp osition If we can recognize these traces we can recursively set up a decomposition

tree that contains structural information ab out a graph

The comp ositiondecomp osition principle that wehave just outlined is the basis for a graph

theoretic approach to the set packing problem The idea of this approachistodevelop algo

rithms that work as follows A given graph is recursively decomp osed into basic comp onents

ie comp onents that can not b e decomp osed further the set packing problem is solved for

each comp onent and the individual solutions are comp osed into an overall solution by going

the decomp osition tree up again

To develop such an algorithm we need the following ingredients A join op eration an e

cient pro cedure that can construct the asso ciated decomp osition tree for a large class of

graphs a metho d to solve the set packing problems at the leafs of the decomp osition tree

and a way to comp ose an optimal stable set in a join from optimal stable sets in comp onent

graphs The last of these four tasks is where polyhedral investigations of joins come into

play Namely if the join op eration is such that one can construct a complete description for

the set packing p olytop e of a join from complete descriptions for its comp onents and such

descriptions are known at the leafs then such a system can also b e constructed for the ro ot

and used to solve the original set packing problem

Our rst example of a join comp oses from two graphs G V E and G V E their

union G G V V E E When the intersection G G V V E E

The Set Packing Polytop e

of the comp onent graphs is a clique this union is called a clique identication Lo oking at

the pro cedure from a decomp osition p oint of view clique identication oers a decomp osition

opp ortunity whenever we can identify in a graph a node separator that is a clique

Theorem Clique Identication Chvatal

Let G G b e a clique identication of two graphs G and G and let A x b and A x b

b e complete descriptions of the set packing p olytop es P G and P G Then

I I

The union of the systems A x b and A x b is a complete description of P G G

Unions of graphs that intersect on a coedge ie on two nonadjacent no des were studied by

Barahona Mahjoub As in the case of clique identication the set packing p olytop es

of coedge identications can also be describ ed completely if such knowledge is available for

the comp onents This technique can b e used to decomp ose a graph that has a co edge no de

separator

Co edge identicationdecomp osition b ears on the derivation of complete descriptions of set

packing polytopes that are asso ciated to W free graphs ie graphs that do not have a

sub division of a wheel as a minor It is known that such graphs can b e decomp osed into a

numb er of comp onents where complete descriptions are known among them sub divisions of

K The decomp osition uses only three typ es of no de separators No de and edge separators

cliques of size one and two and co edge separators Using Chvatal s result on complete

descriptions for clique identications in the rst two and their own result in the co edge case

Barahona Mahjoub construct a p olynomial sized complete extended description of

the set packing p olytop e of a general W free graph G V E Here the term extended

description refers to a system that denes a p olytop e P in a high dimensional space that can

b e pro jected into R to obtain P G extended descriptions take advantage of the observation

that a pro jection of a p olytop e can have more facets than the p olytop e itself

Thelasttypeofjointhatwewanttomention is the amalgamation of two graphs of Burlet

Fonlupt This concept subsumes the graph theoretic op erations of no de substitution

and clique identication it characterizes the class of Meyniel graphs Burlet Fonlupt

show that one can obtain a complete description of the set packing p olytop e of the amalgam

from complete descriptions for the comp onents

Polyhedral Results on Claw Free Graphs

Wehave collected in this subsection some results ab out set packing p olyhedra that are asso

ciated to claw free graphs Most of this material ts into other subsections of this surveybut

the extent of the topic and some unique asp ects seemed to suggest that a treatment in one

place would b e more appropriate

Claw is a synonym for K see Figure and a claw freegraph is one that do es not contain

such a structure

Figure A Claw

Integer Programs

Claw free graphs stir the interest of the p olyhedral community b ecause the set packing prob

lem for this class can b e solved in p olynomial time see Minty but a complete p olyhedral

description is not known The research ob jective is to determine this description

Line graphs are claw free and it was initially susp ected that the facets of the set packing

p olytop es of claw free graphs resemble the facets of the matching p olytop e and would not

b e to o complicated one early conjecture was eg that the only co ecients on the lefthand

side are and Giles Trotter w ere the rst to p oint out that these p olytop es are

complex ob jects They did not only provethe conjecture false but gave also examples

of claw free graphs that pro duce complicated inequalities that contain eg arbitrarily large

co ecients We have mentioned two such classes in Subsection The comp ositions of

circulants and the wedges one can and m ust delete some edges in a wedge as dened in this

thesis to makeitclaw free

Some progress was made by Pulleyblank Shepherd for a the more restrictiv e class of

distance claw free graphs These are graphs that do for eachnodev not only not contain a

stable set of size in the neighborhood of v but they do also not contain such a stable set in

the set of no des that have distance from v Pulleyblank Shepherd give a p olynomial time

dynamic programming algorithm for the set packing problem in distance claw free graphs

and derive a p olynomial sized complete extended description of the asso ciated p olytop e

Gallucio Sassano tak e up the sub ject of general claw free graphs again and investigate

the rank facets that are asso ciated to such graphs It turns out that there are only three

typ es of rank facet producing claw free graphs Cliques line graphs of minimal connected

hyp omatchable graphs and the circulants C All rank facets can b e obtained from

these typ es either by sequential lifting or as sums of two such graphs

We nally mention Euler Le Verge s list of complete descriptions of set packing

p olytop es of claw free graphs with up to no des

Quadratic and Semidenite Relaxations

Next to the search for facet dening and pro ducing graphs the study of quadratic and inti

mately related also of semidenite relaxations is a second general technique to derive valid

and facet dening inequalities for the set packing p olytop e While the rst concept has a

combinatorial and in the rst place descriptive avour the quadraticsemidenite techniques

are algebraic and even b etter algorithmic by their very nature and they do not only apply

to arbitrary integer programs And the metho ds wider scop e and to set packing but

builtin separation machinery is not b ought with a dilution of strength Quite to the contrary

almost all of the explicitly known inequalities for set packing p olytop es can b e pinp ointed in

the quadraticsemidenite setting as well and more Sup erclasses of imp ortanttyp es of con

straints most notably clique and antihole inequalities can b e separated in p olynomial time

This implies in particular one of the most sp ectacular results in combinatorial optimization

The p olynomial time solvability of the stable set problem in p erfect graphs There is only

one price to pay for all of these achievements The number of variables is squared

We try to give in this subsection a survey over some basic asp ects of quadratic and semidenite

techniques for the set packing problem It go es without saying that we can not do more than

scratching the surface of this fast developing eld and we refer the reader to the book of

Grotschel Lovasz Schrijver and the article of Lovasz Schrijver and the

references therein for a more comprehensive treatment Our exp osition is based on the latter

publication and fo cusses on the sp ecial case of the set packing problem

The Set Packing Polytop e

We start byintro ducing the concepts of a quadratic relaxation andasa particularly strong

variant of such a mo del of a semidenite relaxation of the set packing problem in some

graph G V E with n no des that will be numb ered V f ng The idea is to

consider not the convex hull of the incidence vectors x of the stable sets in R but the convex

T nn

hull of the matrices xx R We will see that this quadratic representation gives rise to

two additional cut generation pro cedures that are not available in the linear case

It is technically easier to study quadratic mo dels that gives rise to cones instead of p olyhe

dra and this is the reason to consider a homogenization of the set packing polytope that is

constructed with the aid of an additional comp onent x

f ng

H G cone fgP R

I I

P G can b e retrieved from this ob ject by an intersection with the hyp erplane x We

intro duce also a fractional relaxation of H G that is obtained by replacing P G with

I I

P G and denoted by H G we will assume here and elsewhere in this subsection that P G

is describ ed canonically by the nonnegativity and the edge constraints we assume also that

there are no isolated no des We will work in this subsection only with the cone versions of

the set packing polytope and its fractional relaxation and call H G the set packing cone

and H G the fractional set packing cone Going to quadratic space we are interested in the

set packing matrix cone

T f ngfng

M Gfxx R j x H Gg

I I

The way to construct a quadratic relaxation of the set packing matrix cone is not just to

replace H GwithH G in the denition of the set packing matrix cone whichwould yield

a trivial quadratic relaxation Instead one can set up the following stronger relaxation

T T

QSP i e Ye e Ye i j

j i

i j

T T

ii e Ye e Ye i

i i

iii u Ye u H G i n

u Yf u H G i n

f ngfng

iv Y R

Here we denote by S the polar of a set S and by f i n the vectors of the form

f e e where e is the ith unit vector Their purp ose is to serve as the lefthand sides

i i i

of facets of the homogenized unit cub e U cone fg which contains H G

cone Asso ciated to the system QSP is the fractional set packing matrix cone M G and this

will serve as one relaxation of M G in quadratic space

Before we take a closer lo ok at this ob ject and at the system QSP let us quickly intro duce

another innite set of linear inequalities that strengthens the quadratic relaxation QSP to

a semidenite relaxation that wedenoteby QSP

T f ng

v u Yu u R

Asso ciated to the system QSP is another fractional matrix cone M G

QSP i states that the matrices that are solutions to the system QSP are symmetric a

T fng

prop erty that surely holds for all matrices xx with x H G f g ii states a

typ e of quadratic constraints For matrices xx as ab ove ii stipulates x x i n

i i

Integer Programs

T T

iii throws in what we knowaboutH G Again for matrices xx wehave that u x for

all lefthand sides u H G of valid constraints for H G and since H G U the same is

true for the facets v U of the homogenized unit cub e U which are exactly the vectors e

T T

and f and this yields u xx v iv is the same as stating that the matrix Y is p ositive

semidenite which clearly holds for all matrices of the form xx

The quadratic constraints QSP ii and the semidenite constraints QSP v are not

available in the linear case and account for the greater strength of QSP and QSP in

comparison to the trivial quadratic relaxation One improvement is eg the following Con

sider the vector u e e e H G which is the lefthand side of the edge inequality

i j

x x x x x x for H G and the vector e U which is the lefthand

i j i j j

side of the nonnegativity constraint x forU Inserting these vectors in iii yields

T T

u Ye e e e Y y y y y

j i j j ij jj j ij

and this implies y for all ij E This prop erty do es not hold for the trivial relaxation

But QSP as well as QSP are not only strong they are also algorithmical ly tractable

In fact QSP is of p olynomial size and could be written down easily a prop erty that do es

not hold for QSP but one can solve the separation and the optimization problem for this

system in p olynomial time as well see Grotschel Lovasz Schrijver

Having these ne relaxations in quadratic space at hand wegoback to the original homoge

nized space bya simpleprojection to nally construct go o d relaxations of the homogenized

set packing p olytop e which inherit the sup erior descriptive and algorithmic prop erties of the

matrix cones M GandM G These relaxations are the cones

f ng

N G Me fYe R j Y M Gg

f ng

N GM e fYe R j Y M Gg

and any inequality that is valid for them is a matrix inequality It follows once more from

the general metho dology of Grotschel Lovasz Schrijver that the weak separation

problem for matrix inequalities can be solved in p olynomial time such that one can weakly

optimize over N G and N G in p olynomial time

Theorem N and N Op erator Lovasz Schrijver

Let G b e a graph let H G b e the homogenization of its set packing p olytop e and let H G

b e the fractional edge relaxation of this homogenization Then

H G N G N G H G

The weak separation and optimization problem for N G and N G canbesolved in p oly

nomial time

e remark that the strong separation and optimization problems for N G are also in P W

One can now go one step further and iterate the construction of the matrix cones obtaining

tighter descriptions in every step Inserting N Ginto QSP iii in the place of H G one

obtains a second matrix cone M G that is pro jected into n space to b ecome a cone

N G and so on anyvalid inequality for such a relaxation N G is called a matrix inequality

with N index k Analogously we can iterate the N op erator to obtain relaxations N G

and N matrix inequalities for any natural index k One can show that these relaxations get

tighter and tighter that one can solve the weak separation problem in p olynomial time for

anyxedk and that the nth relaxation coincides with P G I

The Set Packing Polytop e

Theorem Iterated N and N Op erator Lovasz Schrijver

Let G V E b e a graph with n no des let H G b e the homogenization of its set packing

p olytop e and let H G b e the fractional edge relaxation of this homogenization Then

k k

i N G N G k N

k k

ii N G N G k N

iii N G H G

k k

N G can b e solved in p oly The weak separation and optimization problem for N G and

nomial time for every xed natural number k

Theorem gives a wealth of p olynomial time separable classes of inequalities for general

set packing p olytop es namely all matrix cuts with arbitrary but xed N orN index The

graphs whose asso ciated set packing p olytop es can be describ ed completely in this way are

also said to have N orN index k It turns out that a large numb er of graphs have b ounded

indices We rst state the results for the N index

Theorem Graphs with Bounded NIndex Lovasz Schrijver

i An o dd cycle C k has N index

ii A complete graph K has N index n

iii A p erfect graph G has N index G

iv An o dd antihole C k k has N index k

v A minimally imp erfect graph G has N index G

It is more dicult to characterize graphs with b ounded N index but with an analogous def

inition a number of inequalities are known to have small N indices in particular clique and

odd antihole inequalities which are hence contained in the p olynomially separable sup erclass

of matrix inequalities with N index

Theorem Inequalities with Bounded N Index Lovasz Schrijver

Clique o dd cycle wheel and o dd antihole inequalities have N index

As the perfect graphs are exactly those with p erfect clique matrices ie the graphs whose

asso ciated set packing p olytop es can b e describ ed completely by means of clique inequalities

it follows that the set packing problem in p erfect graphs can b e solved in p olynomial time a

sp ectacular result that was rst proved byGrotschel Lovasz Schrijver

Theorem Set Packing Polytop es of Perfect Graphs Grotschel Lovasz

Schrijver Lovasz Schrijver

Perfect graphs have N index

Theorem Set Packing in Perfect Graphs Grotschel Lovasz Schrijver

The set packing problem in p erfect graphs can b e solved in p olynomial time

We nally relate the semidenite relaxation N G to the original approach of Grotschel

Lovasz Schrijver They considered the semidenite formulation QSP but with

iii replaced by

Ye ij E iii e

We denote this semidenite system by QSP and the asso ciated matrix cone by M G

cones cone and this N The pro jection of this matrix cone into n space yields an N

Integer Programs

intersection with the hyp erplane x yields the convex but in general not p olyhedral set

n T T

THG y R j y e Ye i n Y N G fe Ye g

Grotschel Lovasz Schrijver ha ve proved that THG can b e describ ed completely by

means of nonnegativityandorthogonality inequalities Such an orthogonality inequality for

a graph G V E with no des V f ng involves an orthonormal representation of G

n T

ie a set of vectors v R with jv j one for eachnodei of G such that v v holds

i i j

for all ij E and an additional arbitrary vector c R with jcj The orthogonality

inequality that corresp onds to this data is

c v x

i i

This class subsumes the clique inequalities by suitable choices of orthonormal representations

Theorem Orthogonality Inequalities Grotschel Lovasz Schrijver

For anygraphG holds

i Orthogonality inequalities can b e separated in p olynomial time

ii THG is completely describ ed by nonnegativity and orthogonality inequalities

G iii G is p erfect if and only if THGP

Theorem N Index of Orthogonality InequalitiesLovasz Schrijver

Orthogonality inequalities have N index

Adjacency

We summarize in this subsection some results on the adjacency of vertices of the set packing

p olytop e and on the adjacency of integer vertices of its fractional relaxation Such results

b ear on the developmentof primal algorithms for the set packing problem in a graph G

For the purp oses of this subsection we can dene a primal algorithm in terms of a search

graph G V E that has the set of all set packings of G as its no des and some set of edges

set packing to another In every A primal algorithm uses the edges of G to move from one

move the algorithm searches the neighb ors of the current no de for a set packing that has

with resp ect to some given ob jective a b etter value than the current one this neighb orho o d

scan is called a local search When an improving neighbor has been found the algorithm

moves there along an edge of the graph this edge is an improvement direction When there

is no improvement direction the algorithm is trapp ed in a local optimum and stops

The connection between a primal algorithm of lo cal search typ e and the adjacency relation

on a set packing p olytop e P G is that adjacency is a natural candidate to dene the edge

set of the search graph G Namelyweletuv E if and only if the incidence vectors of the set

packings u and v are neighb ors on P G ie if they lie on a common dimensional face of

P G Doing so pro duces a graph GP G which is called the skeleton of P G Skeletons

I I I

have a prop erty that makes them attractive search graphs Not only are they connected but

there is a path of improvement directions from anyvertex to the global optimum

Edmonds famous p olynomial algorithm for set packing problems on line graphs ie

for matching problems moves from one packing to another one by ipping no des in the line

graph of a connected structure that is called a Hungarian tree For maximum cardinality

The Set Packing Polytop e

set packing problems on arbitrary graphs Edmonds has derived a lo cal optimality

criterion that is also in terms of trees and characterizes all improvement directions A set

packing X in a graph G V E is not of maximum cardinalityifand only if the bipartite

graph V X V n X E contains a tree T WF such that X W is a packing of

larger cardinalitythan X Here X Y denotes the symmetric dierence of two sets X and

Y It follows from a result of Chvatal that we will state in a second that Edmondss

matching algorithm do es a lo cal search on the skeleton of the set packing polytope that is

asso ciated to a line graph and that his tree optimality criterion characterizes adjacentvertices

in the skeleton of a general set packing p olytop e In view of these similarities between the

line graphmatching and the general case it was hop ed that matching like primal techniques

could also be applied to general set packing problems An attempt in this direction was

undertaken by Nemhauser Trotter who investigated Edmondss criterion further

and used it supplemented with lower b ounding LP techniques for the development of a

branchandb ound algorithm that could solve maximum cardinality set packing problems on

random graphs with up to no des

The link b etween these results and p olyhedral theory is the following result of Chvatal

that characterizes the adjacent vertices of set packing p olytop es completely The theorem

shows that the ab ovementioned optimality criteria characterize adjacency in the skeleton of

the set packing p olytop e and that the algorithms p erform a lo cal search in this structure

Theorem Adjacency on the Set Packing Polytop e Chvatal

Let G V E b e a graph let P G b e the asso ciated set packing p olytop e and let x and y

b e the incidence vectors of two set packings X and Y in G resp ectively Then

x and y are adjacentonP G if and only if the graph GX Y is connected

Theorem brings up the question if it is p ossible to use p olyhedral information to p erform

a lo cal search in the skeleton One idea to do this was investigated in a series of pap ers by

Balas Padb erg and is as follows Consider the fractional set packing

p olytop e P A that is asso ciated to a given matrix A Any vertex of P A is also a

to reach the integer optimum by searching vertex of P A which means that it is p ossible

through the skeleton GP A of the fractional relaxation This is interesting b ecause there

is an eective and readytouse algorithm that do es exactly this The simplex metho d In

fact nondegenerate pivots lead from one vertex to adjacent vertices and doing additional

degenerate pivots it is p ossible to reach from a given vertex all of its neighb ors In other

words The simplex algorithm p erforms a lo cal search on the skeleton of the fractional set

packing p olytop e with some additional degenerate steps These statements were trivial but

p oint into an interesting direction Is it p erhaps p ossible to move with al l integer pivots

through the skeleton of the integer set packing p olytop e as well It is and GP A seems

to haveeven extremely promising lo oking prop erties

Theorem Skeleton of Set Packing Polytop es Balas Padb erg

Let A be an m n matrix P A the asso ciated set packing p olytop e P A its fractional

relaxation and let G GP A and G GP A b e the asso ciated skeletons Then

I I

i G is a subgraph of G

ii diam G m

iii j vjn v V G

Here diam G denotes the diameter of the graph G

I I

Integer Programs

Theorem i states that vertices are adjacentonP A if and only if they are adjacent

on P A ie it is p ossible to reach the optimum integer solution by a sequence of all integer

pivots And not only is this p ossible By ii one can reach the optimum from any given

integer starting point in at most m pivots The theorem makes only a statement ab out

nondegenerate pivots but one can sharp en this result We remark that Naddef has

proved the Hirsch conjecture true for p olytop es this can sometimes yield a smaller b ound

on the diameter than Theorem ii Finally iii hints to a diculty Eachvertex has

avery large number of neighb ors and this may render the lo cal search dicult

Balas Padb erg havedevelop ed and tested primal pivoting algorithms along these lines

The Set Covering Polytop e

Wesurvey in this section polyhedral results on the set covering p olytop e Q A Analogous to

the set packing case suchinvestigations aim at the characterization of valid and facet den

ing inequalities and the development of metho ds to compute them eciently But the main

motivation for this doing is dierent namely to unify p olyhedral results that were obtained

for various kinds of combinatorial optimization problems that can be stated as optimiza

tion problems over indep endence systems a problem that is in a very direct way equivalent

to the set covering problem F or example the minimal cover inequalities of the knapsack

p olytop e turn out to be socalled generalized clique inequalities and the Mobius ladder in

equalities of the acyclic sub digraph p olytop e can b e seen as generalized cycle inequalities of

an indep endence systemset covering p olytop e that is asso ciated to an appropriately chosen

indep endence system

The concepts that guide these p olyhedral investigations are essentially the same as in the set

packing case One considers facet dening submatrices of a given matrix tries to identify

facet dening classes and uses lifting pro cedures to make lo cal constraints globally valid

The similarity in the approaches carries over to the descriptive resultsandwe will encounter

familiar structures like cliques cycles etc What misses in comparison to set packing are

solvable set covering problems p olynomially separable signicant classes of p olynomially

typ es of inequalities and completely describ ed cases This algorithmic lack is apparently due

to the ineectiveness of graph theoretic approaches to set covering In other words The

algorithmic theory of hyp ergraphs is way b ehind its graphical brother

This section is organized as follows The remainder of this intro duction states some basic

prop erties of the set covering p olytop e most notably the relation to the indep endence system

p olytop e The only Subsection gives a list of facet dening matrices and some results

on rank facets

The subsequent subsections resort to the following basic prop erties of the set covering p oly

alent top e Recall from Section that a set covering problem with matrix A is equiv

to an optimization problem over an independence system IA via an ane transformation

y x

T T T

w ISP max w y SCP min w y

i Ay A I A y

ii y y

iii y y

n n

iv y f y f g g

The Set Covering Polytop e

The indep endence system IA has the set of column indices of A as its ground set and its

cycles are the nonredundantrows of A

The ab ove relation implies that SCP and ISP are equivalent problems and in a very

direct way x is a solution of SCP if and only of x is a solution of ISP This means in

p olyhedral terms that the asso ciated p olytop es satisfy

Q A P A

I ISP

and we need to study only one of them More preciselywehave the following

Corollary Set Covering and Indep endence System Polytop e

Let A b e a matrix and Q A and P A b e the asso ciated set covering and indep endence

I ISP

system p olytop e resp ectively Then

T T T

a x is v alida facet for Q A a x a is valida facet for P A

I ISP

The signicance of the set coveringindep endence system p olytop e for combinatorial opti

mization is that p olyhedral results for Q AP A carry over to manycombinatorial opti

I ISP

mization problems Namelycombinatorial optimization problems can often b e interpreted as

optimization problems over sp ecial indep endence systems and this means that their p olytop es

inherit all facets of the more general b o dy Wewillpoint out some relations of this typ e that

have b een observed in the literature next to the discussion of the corresp onding classes of

inequalities

Some simple prop erties of the set covering p olytop e are collected in

and Nonnegativity Observation Dimension Up Monotonicity Bounds

Let A be a matrix that has at least nonzero entries in each row and Q A be the

asso ciated set covering p olytop e

i Q A is full dimensional

ii Q A is up monotone ie x Q A y Q A for all x y

I I I

iii The upp er b ound constraints x induce facets of Q A

j I

iv A nonnegativity constraint x denes a facet of Q A if and only if the minor A

j I

that results from A by a contraction of column j has at least nonzeros in each row

v If a x denes a facet of Q A that is not one of the upp er b ound constraints x

I j

all co ecients of a x are nonnegative

Facet Dening Matrices

The technique that is used in the literature to derive classes of valid and facet dening

inequalities for the set covering p olytop e is the study of submatrices of a given m n

matrix A similar to the study of subgraphs of the conict graph in the set packing case

Likewise this approach is motivated by and related to the study of minimal ly nonideal matrix

minors The general theory of nonideal matrices guarantees the existence of certain cores

of lo cally facet dening structures

Let us get more precise The derivation techniques for inequalities for set covering p olytop es

from submatrices are based on the upmonotonicityof Q A NamelyifA is some arbitrary

I IJ

minor of A where I is some set of row indices of A and J some set of column indices and

T n

the nonnegative inequality a x is valid for Q A fx R j x j J g and

I I j

Integer Programs

has a for j J it is also valid for Q A The simple extension technique that we

j I

have just describ ed is not very satisfactory but it p oints to the principle that substructures

of A give rise to valid and facet dening inequalities This motivates the concept of a facet

dening matrix in analogy to facet dening graphs for set packing problems We say

that the matrix A denes the facet a x if this inequality is essential for Q A A

rst research topic on set covering p olytop es is now to undertake a classication of such

facet dening matrices and the corresp onding inequalities The facet dening matrices will

serve as candidates for minors of some given matrix of interest Having identied such

a minor we can set up a corresp onding inequality and try to extend it to an inequality that

is globally valid The investigation of p ossibilities to do this extension in a systematic way

leads to the study of lifting techniques The lifting problems for set covering inequalities are

slightly more complicated then in the set packing case b ecause one deals with additional

columns and rows but the general principle is the same we refer the reader to Nemhauser

Trotter Sassano and Nobili Sassano for examples of sequential and

combinatorial simultaneous lifting pro cedures

We give next a list of facet dening matrices for the set covering p olytop e

Generalized Antiweb Inequalities Laurent Sassano For natural

n matrix that numbers n t q a generalized antiweb AW n t q is a n

has a row consecutive column indices e for each q element subset Q of each set of t

fj j t g indices taken mo dulo n see Figure Asso ciated to this matrix is the

generalized antiweb inequality

B C

x dnt q te

Figure The Generalized Antiweb AW

The generalized antiweb inequalityisfacet dening if and only if either n t or t do es not

divide nq see Laurent and Sassano

tiwebs subsume a number of structures that have been investigated earlier Generalized an

Generalized cliques n t by Nemhauser Trotter Sekiguchi and Euler

Junger Reinelt generalized cycles q t and t do es not divide nby Sekiguchi

and Euler Junger Reinelt and generalized antiholes n qt by Euler Junger

Reinelt The last mentioned authors have also investigated some generalizations

of their antiwebs that arise from i duplicating columns of the matrix AW n t q any

numb er of times and ii adding anynumb er of additional columns with certain rather general

prop erties like not having to o many nonzero entries see also Schulz Section for

some further extensions They can show that these generalizations are also facet dening An

application to the indep endence system of acyclic arc sets in a complete digraph exhibits the

classes of k fence inequalities for the acyclic subdigraph polytope as generalized clique and the

Mobius ladder inequalities as generalized cycle inequalities a further example is mentioned

in Nobili Sassano Nemhauser Trotter mention a relation to the knapsack

The Set Covering Polytop e

problem where the class of cover inequalities turns out to corresp ond to the generalized clique

inequalities of the asso ciated indep endence system p olytop e see also Padb erg b

The antiwebs AW k the odd holeshavebeeninvestigated further by Cornuejols

Sassano They study the eects of switching zeros in o dd holes to ones and can com

pletely characterize the cases where such op erations do not destroy the validity and faceteness

of the o dd hole inequality

Sassano and Nobili Sassano give two further and more complicated classes

of facet dening matrices that arise from certain op erations on the antiwebs AW n q q one

a lifting the other a comp osition op eration

Generalized Web Inequalities Sassano Generalized webs are the complements

of generalized antiwebs For natural numbers n t q the generalized web W n t q is a

t n

n matrix that has a row e for each q element subset Q of column n

q q

indices such that Q is not contained in any of the sets fj j t g indices taken

mo dulo nof t consecutive column indices see Figure Asso ciated to sucha web matrix

is the generalized web inequality

B C

x n t

Figure The Generalized Web W

which is facet dening if t do es not divide n see Nobili Sassano

Further Inequalities The inequalities that we have considered so far were all rank in

equalities ie they had all only co ecients on their lefthand sides We mention now

two classes of facets with more general co ecients

Nobili Sassano ha ve studied a class of inequalities from compositions of rank facets

Starting p oint is a matrix op eration the complete bipartite composition that constructs from

two matrices A and A the new matrix

A E

A A

E A

Here E denotes a matrix with all one entries Nobili Sassano sho w that if the rank

T T

inequality x is valid for Q A and the second rank inequality x is valid

A the inequality and in addition tightforQ

T T

x x

denes a facet of Q A A

Finallywemention that Balas Ng ab have completely characterized those facets of

the set covering p olytop e that have only co ecients of and on the lefthand side

Integer Programs

A second branch of research on the set covering p olytop e is devoted to the study of necessary

and sucient conditions that makeavalid inequality facet dening Like in the set packing

case the literature fo cusses on the class of rank inequalities To set up this class consider an

m n matrix A and dene its rank as the number

T n

Amin x Ax x f g

Then the inequality

x A

is the rank inequality that is asso ciated to A Rank inequalities are valid by denition but

there is no complete characterization of those matrices known that give rise to facet dening

rank constraints But a numb er of necessary and sucient conditions have b een derived that

w esurvey next It will turn out that deletion minors play an imp ortant role in this context

and for the remainder of this subsection wewant to denote by A the deletion minor of A

that results from a deletion of all columns that are not contained in J ie A consists of

the columns of A that have indices in J and those rows whose supp ort is contained entirely

in J This matrix is the uncovered part of A that remains when one sets all variables x

j J to

The necessary conditions for a rank inequalityto b e facet dening can be given in terms of

the notions of closedness and nonseparability Wesay that a set J of column indices is closed

if A A holds for all columns k J ie if the addition of any k to J strictly

J fk g

increases the rank J is nonseparable if A A A holds for any partition

J J

J of J into sets J and J ie a separation results in a loss of rank J J

Observation Necessary Conditions for Rank Facets

Let A be an m n matrix let J be any subset of column indices and let A be the

minor that results from deleting from A the columns that are not in J Then

x A denes a facet of Q A the set J must b e closed and If the rank inequality

j I

j J

nonseparable

There are some cases where the condition in Observation is known to b e also sucient

When A is the circuitno de incidence matrix of a matroid and when the indep endence sys

tem IA that is asso ciated to A is the set of solutions of a single knapsack problem see

Laurent

A sucient condition for the faceteness of the rank inequality that is asso ciated to an m n

aph G V E This graph has the set matrix A can b e stated in terms of a critical gr

of column indices of A as its no des and two no des i and j are adjacent if and only if there

n T

exists a covering x Q A Z that satises x A and such that the vector x e e

I i j

which results from exchanging the elements i and j is also a feasible covering

Observation Sucient Condition for Rank Facets Laurent

Let A be an m n matrix let Q A b e the asso ciated set covering p olytop e and let G

b e the critical graph of A Then

If G is connected the rank inequality x A denes a facet of Q A

j I

This observation generalizes a numb er of earlier results of Sekiguchi Euler Junger

Reinelt and Sassano

The Set Covering Polytop e

We close the discussion of rank inequalities for the set covering p olytop e with two approaches

to the heuristic separation of cuts this typ e

Rank Inequalities From KPro jection Nobili Sassano Given an m n

matrix A a subset J f ng of columns its complement J f ngnJ and

an integer k Z a k projection of A with resp ect to J is a matrix Aj with

xJ k

J j columns and the prop erty that anyofitscovers can b e extended to a cover of A n jJ j j

that contains exactly k columns from the set J This matrix is unique up to p ermutation of

rows in fact Aj bl bl Aj where bl Aj is the submatrix of bl A that

xJ k A J k A J k

i J i

has as its rows all the covers of A that contain exactly k columns from J One can prove that

Q Aj is the orthogonal pro jection of the equality constrained set covering p olytop e

xJ k

n J

conv fx f g Ax xJ k g onto the subspace R hence the name k pro jection

The op eration has the prop erty that A Aj k

xJ k

Under sp ecial circumstances k pro jections can b e used to construct rank inequalities Namely

supp ose that the equation A Aj k holds such that A is k projectable with

xJ k

resp ect to J aswesay In this case we can write the rank inequality asso ciated to A as

x Aj k A

xJ k

ie we can construct it from the rank inequality for Aj which is simpler in the sense

xJ k

that it has a smaller righthand side

Nobili Sassano suggest a separation heuristic for rank inequalities that is based on

the iterative application of k pro jections They fo cus on the simplest case where k and J

is the supp ort of a row of the original matrix A ie they always pro ject with resp ect to one of

the equations A x Pro jectability is established using two exp onential sucient criteria

whicharechecked in a heuristic way As the construction of the pro jections is exp onential

as well the authors resort to heuristically chosen submatrices at the cost of a weakening of

the righthand side Pro jection is continued until the resulting matrix b ecomes so small that

the covering number can be determined exactly The separation routine augmented by a

clever lifting heuristic has been successfully used in a branchandcut co de for the solution

of set covering problems from a library of randomly generated instances from the literature

with several hundred rows and columns

Conditional Cuts Balas Balas Ho The cutting planes that we

consider in this paragraph are sp ecial in the sense that they can very well and are indeed

supp osed to cut o parts of the set covering p olytop e under investigation They are only

alid under the condition that a solution that is b etter than the best currently known one v

exists hence the name conditional cut

A more precise description is the following Supp ose that an upp er b ound z on the optimum

ob jectivevalue of the set covering problem SCP is known and consider an arbitrary family

W of column index sets v If we can ensure that the disjunction xv holds

for any solution x with a better objective value than z the inequality

j supp A nv

r v

A such that c xz and can be used as a cutting plane Here for is valid for all x Q

u I

each column set v A is an arbitrary rowof A

r v

Integer Programs

Conditional cuts are of rank typ e the concept subsumes a number of earlier classes such

as the ordinal cuts of Bowman Starr and Bellmore Ratli s cuts from

involutory bases Balas Ho suggest a separation heuristic for conditional cuts that

is based on LP duality arguments the pro cedure has b een applied with success in a branch

andcut algorithm to set covering problems with up to rows and columns

C n P

C n P C n P

C C

C n P C n P

Chapter

Set Packing Relaxations

Summary This chapter is ab out set packing relaxations of combinatorial optimization

problems asso ciated with acyclic digraphs and linear orderings cuts and multicuts multiple

knapsacks set coverings and no de packings themselves Families of inequalities that are valid

for such a relaxation and the asso ciated separation routines carry over to the problems under

investigation

Acknowledgement The results of this chapter are jointwork with Rob ert Weismantel

Intro duction

This chapter is ab out relaxations of some combinatorial optimization problems in the form

of a set packing problem and the use of such relaxations in a p olyhedral approach

Set packing problems are among the b est studied combinatorial optimization problems with

ersons antiblo cking theory the a b eautiful theory connecting this area of research to Fulk

theory of p erfect graphs p erfect and balanced matrices semidenite programming and other

elds see the previous Chapter of this thesis for a survey Likewise the set packing polytope

ie the convex hull of all no de packings of a graph plays a prominent role in p olyhedral com

binatorics not only b ecause large classes of facet dening inequalities are known Perhaps

even more imp ortant many of them can b e separated in p olynomial time in particular o dd

cycle and orthogonality constraints see Grotschel Lovasz Schrijver and Lovasz

Schrijver

OttovonGuericke Universitat Magdeburg Fakultat fur Mathematik Institut fur Mathematische Opti

mierung Universitatsplatz Magdeburg Email robertweismantelmathematiku nim agde burg de

Set Packing Relaxations

Our aim in this chapter is to transfer some of these results to other combinatorial optimization

problems We show that the acyclic sub digraph and the linear ordering problem the max cut

the k multicut and the clique partitioning problem the multiple knapsack problem the set

covering problem and the set packing problem itself haveinteresting combinatorial relaxations

in form of a set packing problem Families of inequalities that are valid for these relaxations

and the asso ciated separation routines carry over to the problems under investigation The

pro cedure is an application of a more general metho d to construct relaxations of combinatorial

optimization problems bymeansof ane transformations

The chapter contains seven sections in addition to this intro duction Section describ es our

metho d to construct set packing relaxations Section is devoted to a study of the acyclic

sub digraph and the linear ordering problem see Grotschel Junger Reinelt ab A

main result here is that a class of Mobius ladders with dicycles of arbitrary lengths b elongs

to a larger class of odd cycles of an appropriate set pac king relaxation this sup erclass

is p olynomial time separable Section deals with set packing relaxations of the clique

partitioning the k multicut and the max cut problem see Grotschel Wakabayashi

and Deza Laurent We intro duce two typ es of inequalities from odd cycles of

triangles The rst of these classes contains the chorded cycle inequalities the second

is related to circulant inequalities Section treats the set packing problem itself We

show in particular that the wheel inequalities of Barahona Mahjoub and Cheng

Cunningham are odd cycle inequalities of a suitable set packing relaxation We also

intro duce a new family of facet dening inequalities for the set packing p olytop e the cycle

of cycles inequalities This class can be separated in p olynomial time Section deals

with the set covering problem Again we suggest a set packing relaxation in order to derive

p olynomially separable inequalities We have implemented one version of such a separation

pro cedure for use in a branchandcut co de for set partitioning problems Implementation

details and computational exp eriences are rep orted in Section of this thesis Section

considers applications to the multiple knapsack problem see Martello Toth and

Ferreira Martin Weismantel The nal Section relates some results of the

literature on set packing relaxations for in teger programming problems with nonnegative

constraint matrices to our setting

The following sections resort to some additional notation and twowell known results for the

set packing or stable set problem Let

SSP max w x

x f g

be an integer programming formulation of a set packing problem on a graph G V Ewith

V E V

nonnegative integer no de weights w Z where A AG f g is the edgeno de

incidence matrix of G Asso ciated to SSP is the set packing or stable set polytope that we

denote in this section by

S n

v S is a stable set in G conv x f g Ax P con

SSP

or where convenient also by P G For reasons that will b ecome clear in the next section

SSP

we will actually not work with the stable set p olytop e P itself but with its antidominant SSP

V V

P P R x R y P x y

SSP SSP SSP

Intro duction

P P

Figure APolyhedron and Its AntiDominant

This construction allows to include vectors with arbitrary negative co ordinates without de

stroying the p olyhedral structure of P Obviously the valid inequalities for P are

SSP SSP

exactly the valid inequalities for P with nonnegative coecients Since the stable set p oly SSP

top e P is down monotone its nontrivial constraints all have nonnegative co ecients and

SSP

wecanthus work with P as well as with P Figure gives an illustration of a p olytop e

SSP SSP

and its antidominant

We will need two results ab out P that are summarized in the following two theorems SSP

Theorem Edge Clique and Odd Cycle Inequalities Padb erg a

Let G V E be a graph and let P be the antidominantof the asso ciated set packing

SSP p olytop e

i If ij is an edge in Gthe edge inequality x x is valid for P

i j SSP

ii If Q is a clique in Gtheclique inequality

b b

P it is facet dening for P if and only if Q is a maximal clique with is valid for

SSP SSP

resp ect to set inclusion

iii If C V is the no de set of an o dd cycle in Gthe o dd cycle inequality

x jC j

P is valid for

SSP

Separation of clique inequalities is NP hard see Garey Johnson Problem GT But

the clique inequalities are contained in the class of orthogonality inequalities see Grotschel

Lovasz Schrijver that can b e separated in p olynomial time Odd cycle inequalities

are p olynomial time separable see again Grotschel Lovasz Schrijver Lemma

Theorem Orthogonality Cycle Inequalities Grotschel et al

Let G V E be a graph P the antidominant of the asso ciated set packing p olytop e

SSP

and x Q Supp ose that x x holds for all edges ij E Then

i j

i Orthogonality inequalities for P can b e separated in p olynomial time

SSP

P can b e separated in p olynomial time ii Odd cycle inequalities for SSP

Set Packing Relaxations

The Construction

Our aim in this section is to describ e a method to construct set packing relaxations of combi

natorial optimization problems The setting is as follows We are interested in some combi

natorial optimization problem that is given byaninteger programming formulation

T n

IP max w x Ax b x Z

mn m n

Here A Z b Z and w Z are an integral matrix and vectors resp ectively The

asso ciated fractional and integer p olyhedra are

n n

bg P A bfx R j Ax bg and P A bconv fx Z j Ax

If the meaning is clear wealsowriteP for P A b and P for P A b

IP IP

Our metho d starts with an ane function

n n

R R x x

n n n

and vector Q note that the image space can b e of given by a rational matrix Q

higher dimension than the preimage We call such functions aggregation schemes or simply

n n

schemes Ascheme is integer if it maps integer p oints to integer p oints ie Z Z or

nn n

equivalently if and are b oth integer ie Z and Z Finally the image P

of a p olyhedron P under the scheme is called the aggregate or if there is no danger of

confusion simply the aggregate of P

Our motivation for studying aggregations is that they give rise to valid inequalities for some

p olyhedron P of interest Namelyif a x is valid for an aggregate P the expansion

T T T

a a a

x x

of this inequality is a constraintinR whichisvalid for the original p olyhedron P

The facial structure of an aggregate is of course in general as complicated as that of the

original p olyhedron But it is often p ossible to nd a relaxation

P P

of the aggregate P that is of a well studied typ e In this case one can resort to known

inequalities for the relaxation P to get an approximate description of the aggregate P and

via expansion a description of a p olyhedral relaxation of the original p olyhedron P

The crucial p oints in this pro cedure are the choice of the aggregation scheme and the iden

tication of a suitable relaxation The subsequent sections resort to the following metho d

to construct set packing relaxations Starting point is the observation that we are inter

ested in combinatorial programs ie optimization problems IP Asso ciated to such

programs are integer p olyhedra P P Restricting attention to likewise integer schemes

n n

ie Z Z recall the ab ove denition the resulting aggregates are integer as well

P P integer P and integer

IP IP IP IP

The next step is to construct a conict graph G V E To do this weneedascheme that

is b ounded from ab oveby one on the p olyhedron P of interest ie

x x P IP

The Acyclic Sub digraph and the Linear Ordering Problem

P G P G

SSP SSP

Figure Constructing a Set Packing Relaxation

Such schemes give rise to a conict graph as follows G has a no de for every comp onent of

the scheme ie V f ng Wedraw an edge uv between two no des if can not attain

its maximum value of one in b oth comp onents simultaneously

uv E x x x P

u v IP

In this case wesay that u and v are in conict The antidominant P G of the conict

SSP

graph is nowaset packing relaxation of the aggregate P in the sense that IP

P G P

SSP IP

Note that it is not p ossible to replace P GwithP G b ecause the scheme can attain

SSP SSP

negativevalues see Figure for an illustration

Once the set packing relaxation P G P is found inequalities and separation routines

SSP

for P G carry over to the p olyhedron P of interest Given some p oint x to b e tested for SSP

memb ership in P we simply i compute x ii solve the separation problem for x

P G and if a separating hyp erplane a x has b een found iii expand it If all and

SSP of these three steps are p olynomial this yields a p olynomial time separation algorithm for a

class of valid inequalities for P namely for the expansions of all p olynomial time separable IP

and p olynomial time expandable inequalities of P G Promising candidates for this are

SSP

P G in particular the o dd cycle and orthogonality constraints for

SSP

The following sections present examples of set packing relaxations for a variety of combina

torial optimization problems

We remark that for convenience of notation we will o ccasionally consider paths cycles di

paths dicycles etc as sets of no des edges or arcs and we will denote edges as wel l as arcs

with the symbols ij and i j the latter symb ol will b e used in cases likei i

The Acyclic Sub digraph and the Linear Ordering Problem

Our aim in this section is to construct a set packing relaxation of the acyclic sub digraph and

the linear ordering problem in a space of exp onential dimension It will turn out that clique

and odd cycle inequalities of this relaxation give rise to and generalize several classes of

inequalities for the acyclic sub digraph and the linear ordering problem namely fence and

Mobius ladder inequalities We suggest Grotschel Junger Reinelt aas a reference for

the ASP see also Go emans Hall and references thereinfor a recen t study of known

classes of inequalities and Grotschel Junger Reinelt b for the LOP

Set Packing Relaxations

The acyclic sub digraph and the linear ordering problem involve a complete digraph D

V Aonn no des with integer weights w on its arcs a A An acyclic arcset in A contains

no dicycle The acyclic subdigraph problem ASP asks for an acyclic arc set with maximum

weight on its arcs Acyclic arc sets that contain for any pair of no des i and j either the

arc ij or the arc ji are called tournaments The linear ordering problem LOP is to nd a

tournament of maximum weight IP formulations for the ASP and the LOP read as follows

X X

w x max w x max

ij ij ij ij

ij A ij A

i x x i j V i j

ij ji

X X

x jC j dicycles C A ii x jC j dicycles C A ii

ij ij

ij C ij C

A A

iii x f g iii x f g

LOP ASP

ASP is a relaxation of LOP and what is more the linear ordering polytope P is a face

LOP

of the acyclic subdigraph polytope P In particular all inequalities that are valid for P

ASP ASP

arealsovalid for P Two such classes of inequalities for b oth the ASP and the LOP are

LOP

the k fence and the Mobius ladder inequalities see Grotschel Junger Reinelt a

u u u u

C C

l l l l

C C

Figure A Fence

Figure AMobius Ladder of Dicycles

A simple k fence involves two disjoint sets of upp er and lower no des fu u g and

fl l g that are joined by a set of k pales fu l u l g All pales are oriented down

k k k

ward The k fence is completed by adding all upward pickets l u with the exception of the

i j

antiparallel pales We remark that one can also allow that pales and pickets consist not only

of a single arc but of an entire dipath thereby obtaining a larger class of general k fences

for simplicity of exp osition however wewant to restrict ourselves here and elsewhere in this

section to simple fences Figure shows a simple fence

A Mobius ladder consists of an o dd number k of dicycles C C suchthatC and

C indices taken mo dulo k have a dipath P in common see Figure this time

i i

wewant to consider also the non simple case

Fences and Mobius ladders give rise to valid inequalities for P For a k fence F and a

ASP

Mobius ladder M of k dicycles wehave

k k

X X X

and

x k k

x jC n P j k

ij i i

ij F

i i

ij C nP

i i

The Acyclic Sub digraph and the Linear Ordering Problem

Note that a Mobius ladder inequality as ab ove has co ecients larger than one if an arc is

contained in more than one of the dipaths C n P This is dierentfromGrotschel Junger

i i

Reinelt as original denition where the co ecients take only values of zero and one

and the Mobius ladder must meet anumb er of additional technical requirements to supp ort

a valid inequality The two denitions of a Mobius ladder inequality coincide if and only if

no two dipaths C n P have an arc in common Mobius ladder without arc repetition

i i

Wewillshownow that fences and Mobius ladders are cliques and o dd cycles resp ectivelyin

onict graph GD V E G has the set of all acyclic arc sets of D an exp onential c

n n

as its no des We draw an edge uv between two acyclic arcset no des u and v if their union

contains a dicycle In this case wesay that u and v are in conict b ecause they can not be

simultaneously contained in the supp ort of a solution to ASP

u u u u u u u u

u u u u

l l

F F

u u u u u u u u

l l l l

l l

F F

Figure AFence Clique

It is now easy to identify the fences and Mobius ladders with cliques and odd cycles of G

To obtain a k fence F we lo ok at the k acyclic arc sets F that consist of a pale u l and

i i

and the pickets l u that go up from l for i k Any two such congurations F

i j i

i j are in conict they contain a dicycle Hence all of them together form a F clique

Figure illustrates this construction Likewise the Mobius ladders corresp ond to o dd cycles

of conicting dipaths namely the dipaths C n P see Figure

i i

C n P

C n P C n P

C C

C n P C n P

Figure AMobius Cycle of Dipaths

Set Packing Relaxations

The next step to obtain the fence and the Mobius ladder inequalities from the clique and o dd

P G asso ciated to cycle inequalities of the antidominant of the set packing p olytop e

SSP

the conict graph G is to construct a set packing relaxation of the ASP To this purp ose

A V

consider the aggregation scheme R R dened as

x x jvj acyclic arc sets v V

v ij

ij v

xisintegral for all integral x R Moreover for every incidence vector x P of an

ASP

acyclic arc set suppx A in D we have that x attains its maximum value of one in

comp onent x if and only if v is contained in suppx Since two conicting acyclic arc

sets can not simultaneously b e contained in suppx wehave that

uv E x x x P Z

u v ASP

and by convexity also for all x P This argumentproves that P G is a set packing

ASP SSP

relaxation of P

ASP b

GD P Lemma Set Packing Relaxation of the ASP P

n SSP ASP

Note that it is not p ossible to replace P with P b ecause the comp onents of can

SSP SSP

take negative values More precisely x is in general not the incidence vector of a stable

set in P G but max fxg is with the maximum taken in every comp onent recall

SSP

Figure T

Lemma allows us to expand see the denition on page an inequality a x that

is valid for P into the inequality a x for P Our next theorem states that

SSP ASP

the fence and Mobius ladder inequalities are expansions of clique and o dd cycle inequalities

resp ectively

Theorem Fence and Mobius Ladder Inequalities

Let D be the complete digraph on n no des P the corresp onding acyclic sub digraph

n ASP

p olytop e G the conict graph asso ciated to D and P G the set packing relaxation of

n SSP

P ASP

P G i Every k fence inequalityfor P is the expansion of a clique inequalityfor

SSP ASP

ii Every Mobius ladder inequalityfor P is the expansion of an o dd cycle inequality for

ASP

P G

SSP

Pro of

i k dened on the previous page i Let F be a k fence The acyclic arc sets F

form a clique in G see the discussion on the previous page An expansion of the corresp onding

clique inequality yields the desired k fence inequality

k k

X X X X X

A A

x jF j x k x k k

ij ij ij

i i

ij F

ij F ij F

k k

x k k

ij F k

The Acyclic Sub digraph and the Linear Ordering Problem

ii Let M beaMobius ladder consisting of an o dd number k of dicycles C C such

that C and C have a dipath P in common The argument on page showed that the

i i i

dipaths C n P form an o dd cycle of k acyclic arc sets in G Expanding the corresp onding

i i

o dd cycle inequalityfor P G one obtains the Mobius ladder inequalityfor M

SSP

x k

C nP

i i

X X

k x jC n P j

ij i i

ij C nP

i i

k k

X X X

jC n P j k x

i i ij

i i

ij C nP

i i

ladder inequalities have been discussed in a number of dierent frame Fence and Mobius

works in the literature Euler Junger Reinelt interpret fences and Mobius ladders

without arc rep etitions as generalized cliques and generalized odd cycles of an independence

system relaxation of the ASPMuller Schulz giv e cutting plane pro ofs of fence

and Mobius ladder inequalities in the context of transitive packing see also Schulz

Chapter Caprara Fischetti give a derivation of Mobius ladder inequalities in

terms of f g ChvatalGomory cuts The last two constructions work for Mobius ladders

with arc rep etitions and yield a class that is in the middle between Grotschel Junger

Reinelt as Mobius ladder inequalities and ours dep ending on the number of dipaths

that contain a given rep eated arc

Separation of fence inequalities was shown to be NP hard by Muller Lo oking at the

separation of Mobius ladder inequalitieswe notice that the construction that we presented to

prove Theorem ii yields a class of odd cycle of dipath ine qualities that subsumes the

Mobius ladder inequalities Generalizing this class further by allowing the paths C n P to

i i

intersect themselves on no des andor arcs ie by substituting in the denition of a Mobius

ladder on page diwalk for dipath and closed diwalk for dicycle we obtain an even larger

class of odd cycle of diwalk inequalities for the acyclic sub digraph p olytop e Note that these

inequalities do in general not corresp ond to odd cycles in the acyclic arc set conict graph

G b ecause diwalks may contain dicycles This obstacle can be overcome by extending G in

At this p oint however we do not want to a nite way including certain relevant diwalks

enter this formalism and defer the details of the extension to the pro of of Theorem

We can devise a p olynomial time separation algorithm for odd cycle of diwalk inequalities

even though the number of diwalks is exp onential and their length is arbitrary The idea is to

construct a most violated cycle of diwalks out of prop erly interlinked longest diwalks Supp ose

that M is an o dd cycle of diwalks wewant to denote these diwalks with a slight extension of

notation by C n P that induces a violated inequality and consider the diwalk P linking the

i i i

two successive closed diwalks C and C Rearranging we can isolate the contribution of

i i

P in the constraintas

X X X X

x jC n P P j k x jC n P j x jP j

ij i i i ij j j ij i

ij P

j i

ij C nP P ij C nP

i i i j j

Here all sets are to be understo o d as multisets Note also that we have b ecause the

constraintwas by assumption violated

Set Packing Relaxations

If we replace P by a diwalk P that is shorter with resp ect to the length function

X X

x x jP j

ij ij

ij P ij P

we get a more violated cycle of diwalks inequality If we think of any closed diwalk C as b eing

comp osed out of four diwalks namely the diwalk P P that C has in common with the

i i

succeeding closed diwalk C thediwalk P from P s head to the diwalk P P that

i i

i i i

C has in common with the preceeding closed diwalk C and the remaining diwalk P from

i i

P s head to P s tail the same argument holds for any of these diwalks This observation

i i

allows us to show

Theorem Polynomial Separability of Odd Cycle of Diwalk Inequalities

Let D be the complete digraph on n no des and P the asso ciated acyclic sub digraph

n ASP

p olytop e Supp ose that x Q satises the constraints ASP ii and x Then

Odd cycle of diwalk inequalities can b e separated in p olynomial time

Pro of

Using Dijkstras algorithm we can compute a shortest diwalk P u v with resp ect to the

length from any no de u to any no de v of D We can assume these diwalks P u v

wlog to b e dipaths and in particular to b e of p olynomial length This yields a p olynomial

number of shortest diwalks of p olynomial length and moreover not every but a most

violated cycle of diwalks will consist of a p olynomial numb er of these shortest diwalks

We can nd a set of them forming an o dd cycle of diwalks as follows We think of all diwalks

P u v as a p ossible common diwalk P of two successive closed diwalks C and C in a

i i i

cycle of diwalks Togetthediwalks C n P as the pieces of the cycle we compute for anytwo

i i

diwalks P and P the shortest diw alk P hP i that starts at P s head contains P and

i j i j i j

ends at P s tail Such a diwalk P hP i will link on P prop erly with another diwalk P hP i

i i j j j

to form a cycle of diwalks Computation of the P hP i can b e p erformed in p olynomial time

i j

and yields in particular a p olynomial number of nn nn O n diwalks of

p olynomial length

We can now construct a graph that has these diwalks P hP i as its no des with no de weights

i j

hP i The no de equal to the walk lengths and that has all edges of the form P hP iP

i j j

weight onanedgenever exceeds one b ecause x satises the dicycle inequalities ASP ii a

most violated cycle of diwalks inequality corresp onds to a most violated o dd cycle inequality

in the P hP igraph and we can nd a most violated odd cycle inequality there with the

i j

algorithm of Grotschel Lovasz Schrijver Lemma

Corollary Separation of Mobius Ladder Inequalities

A sup erclass of the Mobius ladder inequalities can b e separated in p olynomial time

To discuss the results on Mobius ladder separation of the literature we draw the readers

attention to a subtle dierence between the ASP and the LOP While the length of the

dicycles in a facet dening Mobius ladder inequality as dened in this pap er for the acyclic

subdigraph polytope can be arbitrarily large the constraint can only dene a facet for the

linear ordering polytope if the length of each dicycle is at most four see Grotschel Junger

Reinelt a For the LOP one can thus restrict Corollary to the case jC j and

then it also follows from Muller Schulz and Caprara Fisc hetti For the ASP

Caprara Fischetti sho wed p olynomial time separabilityofMobius ladder inequalities

where all dicycles have at most constantlength

The Clique Partitioning Multi and Max Cut Problem

In this section weinvestigate set packing relaxations of combinatorial optimization problems

in connection with cuts The clique partitioning the k multicut and the max cut problem

We will see that the chorded cycle inequalities for the clique partitioning p olytop e can be

seen as cycles of lower triangle inequalities An analogous construction for cycles of upp er

triangle inequalities is related to the circulant inequalities for the max cut p olytop e As a

reference for the clique partitioning problem we suggest Grotschel Wakabayashi see

also Oosten Rutten Spieksma for a recen t rep ort for the multicut and the max cut

problem Deza Laurent

The three cut problems of this section come up on a complete graph K V Eonn no des

with integer weights w E Z on the edges The clique partitioning problem CPP is to nd

a partition of V into an arbitrary num ber k of cliques V C C where denotes

a union of disjoint sets such that the sum of the weights of the edges that run between

dierent cliques is maximal In other words we are trying to nd a multicut C C

of maximum weight where the number k of non empty members C of the clique partition

C C is arbitrary One obtains the k multicut problem k MCP from this formulation

by restricting the numb er of cliques to b e less than or equal to some given number k and the

max cut problem MCP by prescribing k Thus anymax cut is a k multicut k

and any k multicut comes from a clique partition We remark that the CPP is often stated

in an equivalentversion to nd a clique partition that minimizes the sum of the edge weights

inside the cliques

Integer programming formulations of the clique partitioning and the k multicut problem read

as follows x indicates that ij is in the multicut

max w x

ij ij

max w x

ij ij

ij E

x jE W j W V i

W ij E

with jW j k

ii x x x fi j k gV

jk ik

ii x x x fi j k gV

jk ik

iii x f g

CPP

k MCP

Setting k to inequalities k MCP i turn out to be the upp er triangle inequalities

x x x for all fi j k gV and MCP is an integer programming formulation

jk ik

for the max cut problem we sp eak of upp er triangle inequalities b ecause their normal vectors

are oriented upward such that the induced face is on the upside of the p olytop e For

k n on the other hand k MCP i b ecomes void and nMCP coincides with CPP

Hence CPP is a relaxation of k MCP which in turn is a relaxation of MCP and the

asso ciated p olytop es P P and P satisfy

CPP MCP

k MCP

P P P

CPP MCP

k MCP

In particular any valid inequality for the clique partitioning p olytop e is also valid for the

k multicut and the max cut p olytop e One such class is the family of chorded cycle inequal

ities of Gr otschel Wakabayashi

Set Packing Relaxations

A chorded cycle is an o dd cycle C of K together with its set of chords C see Figure

The asso ciated inequality states that

X X

x jC j x

ij ij

ij C

Figure A Chorded Cycle

Muller and later Caprara Fischetti proved that sup erclasses of the chorded

cycle inequalities can b e separated in p olynomial time see also Muller Schulz We

will show now that this class arises from odd cycle inequalities of a set packing relaxation

of the clique partitioning or k multicut or max cut problem Our arguments will yield an

alternative pro of for the p olynomial time separability of this class

The relaxation involves a lower triangle conict graph G K V E V consists

E of G are of the form of all ordered triples i j k V of distinct no des of K the edges

i j k l i j i j k l j i i j k l i k and i j k l k i the meaning of this denition

will b ecome clear in a second

i i i i i i

k k k k k k

j j j j j j

i j k j i k k j i k i j j k i i k j

Figure Lab eling Lower Triangles

To construct a set packing relaxation of the clique partitioning problem with this graph we

E V

dene an aggregation scheme R R as

xx x ordered triples i j k V

ijk

x is integral if x R is integral Moreover for every multicut x P the

CPP

ijk

j and k x attains its maximum value of one if and only if the no des comp onent

ijk

b elong to the same clique x but no de i do es not x x The reader may

jk ik

think of the triples i j k as edgelab eled triangles as shown in Figure Enumerating

all p ossible conicts b etween these lab eled triangles it is easy to see that

x x P Z x uv E

CPP

v u

and thus for all x P In other words E was dened in suchaway that two triples are

CPP

joined by an edge if and only if it is imp ossible that b oth triples attain their maximum value

of one under simultaneously This argument shows that P G is a lower triangle set

SSP

packing relaxation of P

CPP

G K Lemma Set Packing Relaxation of the CPP P P

n CPP SSP

The Clique Partitioning Multi and Max Cut Problem

The construction is called a lower triangle set packing relaxation because one obtains the

comp onents xx x of from the lower triangle inequalities CPP ii

ijk

x x x x x x

ij ij

jk ik jk ik

by setting x ik

We are now ready to state our result that the chorded cycle inequalities are expansions of

o dd cycle inequalities of P G

SSP

Theorem Chorded Cycle Inequalities

Let K b e the complete graph on n no des P the corresp onding clique partitioning p oly

n CPP

top e G the lower triangle conict graph and P G the lower triangle set packing

SSP

relaxation of P Then CPP

Every chorded cycle inequality for P is the expansion of an odd cycle inequality for

CPP

P G

SSP

Figure An OddCycle of Lower Triangles

Pro of

Let C C be a chorded cycle in K with no de set f k g By denition C fij i

k j i g and C fij i k j i g where indices are taken mo dulo

Consider the k triples v i i i i k indices mo dulo k One

veries that v v E represents a conict and forms an edge of an odd cycle in G see

i i

Figure for an example The asso ciated odd cycle inequality expands to the chorded

cycle inequality in question

k k

X X X X

x x x x x jC j

ij ij

ii ii

iii

i i

ij C

Set Packing Relaxations

Calling the expansions of o dd cycle inequalities for P G inequalities from odd cycles of

SSP

lower triangle inequalities and noting jV j O n we obtain

Corollary Separation of Ineqs from Odd Cycles of Lower Triangle Ineqs

Let K b e the complete graph on n no des and P the asso ciated clique partitioning p oly

n CPP

top e Supp ose x Q satises the constraints CPP ii and x Then

Inequalities from o dd cycles of lower triangle inequalities can b e separated in p olynomial time

Corollary Separation of Chorded Cycle Inequalities

A sup erclass of the chorded cycle inequalities can b e separated in p olynomial time

i Note that the conicts b etween successive triples v i i i and v i i

i i

in a chorded cycle stem from the common edge connecting no des i and i that has a

co ecient of in and in But conicts arise also from common edges with

v v

i i

and co ecients Thus b esides p ossible no deedge rep etitions and the like o dd cycles of

lower triangle inequalities give also rise to inequalities that do not corresp ond to chorded

cycle inequalities

So far we have studied inequalities from pairwise conicts of lower triangle inequalities In

the case of the max cut problem the constraints MCP i form a class of upp er triangle

inequalities

i j k V x x x

ik jk

Analogous to the lower triangle case we will now construct inequalities from o dd cycles of

upp er triangle inequalities for the max cut p olytop e These constraints are related to the

C k circulant inequalities of Poljak Turzik

A C k circulant is identical to a chorded cycle on an o dd number of k no des

We distinguish circulants C k with odd k and with even k The asso ciated inequalities

are

X X

x k if k x k if k mo d mo d

ij ij

ij C k ij C k

C C

Figure The Oddk Circulant C Figure The Evenk Circulant C

Evenk circulant inequalities have been intro duced by Poljak Turzik These con

straints havearighthand side of k whereas the o dd case requires an increase of one in the

The Clique Partitioning Multi and Max Cut Problem

righthand side Figure shows the oddk circulant C k the white and gray

no des form the shores of a cut with edges highlighted Alternatingly

putting pairs of successive no des on the left and on the right shore of the cut except for the

rst no de one veries that the righthand side k is b est p ossible for o dd k Figure

shows a tight conguration for the case where k is even A rigorous pro of for the validityof

these constraints will follow from the up coming discussion

We will show now that the circulant inequalities can be seen as strengthened odd cycle

inequalities of an appropriate upp er triangle set packing relaxation of the max cut prob

lem strengthened means that for even k the righthand side of the cycle of upp er triangles

inequality exceeds the righthand side of the corresp onding circulant inequalityby one Our

considerations allow to design a p olynomial time algorithm for separating inequalities from

cycles of upp er triangle inequalities Poljak Turzik on the other hand have shown

that separation of the exact class of C k circulant inequalities is NP hard

i i i

k k k

j j j

j ik i j k k ij

Figure Lab eling Upp er Triangles

As usual the upp er triangle set packing relaxation is based on an upp er triangle conict

graph G K V E This time V consists of all tuples i j k V E such that

i jk while E is the set of all i j k j k l To construct a set packing relaxation of P

MCP

E V

by means of this graph weintro duce the aggregation scheme R R dened as

xx x

ij k

One gets x from the upp er triangle inequality

ij k

x x x x x x

ij ij

ik jk ik jk

by setting x hence the name upp er triangle conict graph This rearrangement also

proves that x attains its maximum value of one if and only if no de i is on one side of

ij k

the cut while no des j and k are on the other Again one may think of the no des i j and k

as forming triangles with the edges lab eled as indicated in Figure and sees that

uv E x x x P Z

MCP

u v

Thus P G is an upp er triangle set packing relaxation of P

SSP MCP

Lemma Set Packing Relaxation of the MCP P P G K

MCP SSP n

This construction yields the circulant inequalities for k mo d as expansions of o dd cycle

inequalities for the upp er triangle set packing relaxation of the max cut p olytop e The case

k mo d can b e settled by strengthening the asso ciated o dd cycle inequality ie one can

a posteriori decrease the righthand side by one and the inequality remains valid

Set Packing Relaxations

Theorem Circulant Inequalities

Let K be the complete graph on n no des P the corresp onding max cut p olytop e G

n MCP

the upp er triangle conict graph and P G the upp er triangle set packing relaxation of

SSP

P MCP

i Every o ddk circulant inequalityfor P is the expansion of an odd cycle inequality

MCP

P G for SSP

ii Every evenk circulant inequalityfor P is the expansion of a strengthened o dd cycle

MCP

inequalityfor P G

SSP

Figure An Odd Cycle of Upp er Triangles

Pro of

i Let C k b e an o ddk circulant with no de set f k g Consider the k

i i i with indices taken mo dulo k One veries that the tuples tuples v

v and v are in conict ie v v E and form an o dd cycle in G see Figure

i i i i

The asso ciated o dd cycle inequality expands to an o ddk circulant inequality

x k

iii

x x k

ii ii

x k

ij C k

The Set Packing Problem

ii The even case is analogous to the o dd case To see that one can reduce the righthand side

x k is still valid supp ose this of the o dd cycle inequalityby one and

iii

is not so and let x P b e the incidence vector of a cut that violates this constraint Now

MCP

max f xg is the incidence vector of a stable set in G and clearlythisvector must b e

tight for the unstrengthened o dd cycle inequality This means that wehave k tuples with

g that are max f xg and k tuples with max f x

iii iii

arranged in suchaway that the twotyp es app ear alternatingly except for one time where we

xg have two zeros next to each other Lo oking at a tuple with max f

iii

we see that no de i must b e on one side of the cut while no des i and i must b e on the

other The next one max f xg forces no des i and i to be on

iii

andcontinuing k times the same side as i Starting without loss of generalityat

like this all no des of the circulantare assigned to one side of the cut or another in a unique

way When k is even this results in no des k k and ending up on the same side

xg see the right side of Figure for an example but then such that

k k

x k a contradiction

iii

Calling the expansion of an o dd cycle inequalityfor P G an inequality from an odd cycle

SSP

of upper triangle inequalitiesweobtain

Corollary Separation of Ineqs from Cycles of Upp er Triangle Ineqs

Let K b e the complete graph on n no des and P the asso ciated max cut p olytop e Supp ose

n MCP

x Q satises the constraints MCP i ii and x Then

Inequalities from o dd cycles of upp er triangle inequalities can b e separated in p olynomial time

Corollary Separation of Circulant Inequalities

i A sup erclass of o ddkCk circulant inequalities can b e separated in p olynomial

time

ii A sup erclass of evenk C k circulant inequalities with their righthand sides

increased by one can b e separated in p olynomial time

We remark again that Poljak Turzik ha veshown that it is NP complete to determine

whether a graph contains a C k circulant and thus separation of the exact class of

C k circulant inequalities is NP hard

The Set Packing Problem

Wehave demonstrated in the examples of the preceeding sections that certain combinatorial

optimization problems have interesting set packing relaxations Perhaps a bit surprising we

show now that the set packing problem itself also has interesting set packing relaxations

These considerations yield alternative derivations generalizations and separation techniques

for several classes of wheel inequalities including two classes intro duced by Barahona

Mahjoub and Cheng Cunningham as w ell as new classes suc h as eg certain

cycle of cycles inequalities A survey on results for the set packing problem can be found

in Chapter of this thesis

The examples of this section are based on a rank set packing relaxation that weintro duce

now Given a set packing problem SSP on a graph G V E the asso ciated conict graph

G V E of the relaxation has the set V fH H Gg of all not necessarily no de

Set Packing Relaxations

induced subgraphs of G as its no des In order to dene the set of edges we consider the

V V

aggregation scheme R R dened as

x x H subgraphs H V of G

H i

iV H

where H denotes the rank ie the maximum cardinality of a stable set of H We drawan

edge b etween two subgraphs H and W if there is no stable set in G such that its restrictions

to H and W are simultaneously stable sets of maximum cardinalityin H and W ie

x x x P G Z HW E

H W SSP

By denition the rank conict graph G dep ends only on G and this is whywe o ccasionally

P G is a set packing relaxation also denote it by GG Well known arguments show that

SSP

of P in the exp onential space R SSP

Lemma Rank Set Packing Relaxation of the SSP P P G

SSP SSP

Wheel Inequalities

One metho d to derive classes of p olynomial time separable inequalities from the rank relax

ation is to consider subgraphs of G of p olynomial size A natural idea is to restrict the set of

Gs no des to

V fH H G jV H jk g

the subgraphs H of G with b ounded numb ers of no des jV H j k for some arbitrary but

xed b ound k The smallest interesting case is k where H jV H j is either empty

a singleton an edge or a co edge complement of an edge The odd cycle inequalities that

one obtains from this restricted relaxation P GV contain among other classes the odd

SSP

wheel inequalities of the set packing p olytop e

Ak wheel is an o dd cycle C of k no des f k g plus an additional no de k

that is connected to all no des of the cycle C is the rim of the wheel no de k is the hub

and the edges connecting the no de k and i i k are called spokes For sucha

conguration wehavethat

kx x k

Figure A Wheel

An o dd wheel inequality can b e obtained by a sequential lifting of the hub into the o dd cycle

inequality that corresp onds to the rim Trying all p ossible hubs this yields a p olynomial time

separation algorithm for wheel inequalities An alternative derivation is

The Set Packing Problem

Figure A Wheel and a Cycle of No des and Edges

Theorem Odd Wheel Inequalities

Let G V E b e a graph P the corresp onding set packing p olytop e G the rank conict

SSP

graph and P G the rank set packing relaxation of P Then

SSP SSP

Every o dd wheel inequalityfor P is the expansion of an o dd cycle inequalityfor P GV

SSP SSP

Pro of

Consider a k wheel with rim C f k g and hub no de k The subgraphs

v Gfi k g i k induced by the sp okes with o dd rim no des and the

subgraphs v Gfig i k induced by the even rim no des form an o dd cycle in

G see Figure Expanding the asso ciated o dd cycle inequality yields the wheel inequality

k k

X X X X

x x x kx x k

i i i

k k

v x

i i

i k ik

Corollary Separation of Ineqs from Odd Cycles of No des Edges Co edges

Ineqs from o dd cycles of no des edges and co edges can b e separated in p olynomial time

We show now two examples of cycles of no des edges and co edges that give rise to facetial

inequalities that do not corresp ond to odd wheels The cycle on the left side of Figure

the edges and the one on the right of the consists of the no des and and

edges and and the co edge The asso ciated inequalities are

x x x x x x x x

x x x x x x x x x x x

X X

x x

i i

Figure Two Generalizations of Odd Wheel Inequalities

Another generalization of o dd wheel inequalities was given by Barahona Mahjoub and

Cheng Cunningham They intro duce two classes of inequalities that have subdivisions

Set Packing Relaxations

of o dd wheels as supp ort graphs where each face cycle must b e o dd see Figure Following

Cheng Cunningham s terminology and denoting the set of end no des of the even

spokes with an even number of edges of an odd wheel W of this kind with some number

k of faces by E the set of end no des of the odd spokes with an odd number of edges

by O and the hub by hawheel inequality of type I states that

X X

jW j jE j

x x kx

i i

iW h

A second variantof wheel inequalities of type II asso ciated to the same wheel states that

X X

jW j jO j

x x k x

i i

iO iW h

We remark that these wheels do in general not arise from cycles of subgraphs of b ounded size

b ecause they contain p otentially very long paths

Theorem Odd Wheel Inequalities

Let G V E b e a graph P the corresp onding set packing p olytop e G the rank conict

SSP

graph and P G the rank set packing relaxation of P Then

SSP SSP

Every o dd wheel inequalityoftyp e I and I I for P is the expansion of an o dd cycle inequality

SSP

for P G

SSP

hub h

even sp oke ends E f g

odd spokeends O f g

Figure A Wheel and a Cycle of Paths of Typ e I

Pro of

i Wheel inequalities of typ e I

The idea of the pro of is to obtain the wheel inequality of typ e I as a cycle of paths

Orienting a k wheel clo ckwise it consists of k sp oke paths S i k and the

same numb er of rim paths R such that R connects the ends of sp okes S and S indices

i i i i

in the pro of are taken mo dulo k We can then comp ose the wheel from the paths

R if S is even if i is o dd

i i

P S n i k

i i

R n S if S is o dd fhg if i is even

i i i

The Set Packing Problem

see Figure By denition a path P consists of the sp oke S plus minus the hub dep ending

i i

on i and the ful l rim path R if the end no de of the next sp oke in clo ckwise order is even

or the rim path R without the end of the next sp oke in case this sp oke is o dd In this way

the even sp okeends having a co ecientoftwo in the wheel inequality app ear in two paths

the odd sp okeends in one Finally the hub is removed from all paths with even index It

is not hard to see that any two successive paths P and P are in pairwise conict The

i i

subpaths P nfhg with the hub removed are all o dd and in pairwise conict and the hub is

in conict with any of these subpaths The o dd cycle inequality corresp onding to the paths P

expands into the o dd wheel inequality

x k

k k

X X X X

A A

x jP j k x jP j

j i j i

i i

j P j P

i i

X X

jW jk jE j k k

x x k kx

j j

j E

j W nfhg

X X

jW j jE j k jW j jE j

kx x x k

j j

j E

j W nfhg

Here jP j denotes the numb er of no des in the path P

i i

hub h

even sp okeends E f g

odd spokeends O f g

Figure A Wheel and a Cycle of Paths of Typ e I I

ii Wheel inequalities of typ e I I

The wheel inequalities of typ e I I can b e derived in much the same way as their relatives

of typeIFor the sake of completeness we record the path decomp osition

R if S is o dd if i is even

i i

P S n i k

i i

R n S if S is even fhg if i is o dd

i i i

Set Packing Relaxations

One can verify that again any two successive paths are in conict A nal calculation to

expand the resulting o dd cycle inequality yields the wheel inequality of typ e I I

x k

k k

X X X X

A A

k x jP j x jP j

j i j i

i i

j P j P

i i

X X

jW jk jO j k k

k k x x x

j j

j O

j W nfhg

X X

jW j jO j k jW j jO j

x x k x k

j j

j O

j W nfhg

One can also derive p olynomial time separation algorithms of much the same avour as for

the Mobius ladder inequalities such pro cedures are given in Cheng Cunningham

A New Family of Facets for the Set Packing Polytop e

The rank relaxation of the set packing problem oers ample p ossibilities to dene new classes

of p olynomially separable inequalities for the set packing problem We discuss as one such

example a cycle of cycles inequality cycleofcliques inequalities and certain liftings of them

are studied in Tesch Section

The way to construct a cycle of cycles inequality is to link an o dd number k of o dd cycles

C C to a circular structure such that anytwo successive cycles are in pairwise conict

ie x x indices taken mo dulo k

C C

i i

One way to do this is to select from each cycle C three successive no des L C that will

i i i

serve as a part of the intercycle links yet to be formed The link L has the prop erty that

x implies that at least one of the no des in L is contained in the stable set suppx

C i

X X

x x jC j x

j j i C

j L j C

i i

If we make sure that any two successive links L and L are joined by the edge set of the

i i

complete bipartite graph K wehavethat

X X

x x x x xP G Z

C j j C SSP

i i

j L j L

i i

and vice versa that x x holds for all incidence vectors x of

C C

i i

stable sets in G But then every two successive cycles C and C are in conict ie

i i

C C

i i

x and the cycles C form an o dd cycle in G see Figure links are x

colored gray

The Set Packing Problem

C L

Figure A Cycle of Cycles

Theorem Cycle of Cycles Inequality

Let G V E be a graph and P be the corresp onding set packing p olytop e Let C

i SSP

C i k a set of three successivenodes i k b e an o dd cycle in G and L

i i

in C Assume further that L and L i k are joined by a complete K Then

i i i

The cycle of cycles inequality

k k

X X X

C j k x j

i j

i i

j C

is valid for P

SSP

Pro of

x k

X X

k x jC j

j i

j C

k k

X X X

jC j k x jC j k

i j i

i i

j C

Theorem Separation of Cycle Of Cycles Inequalities

Cycle of cycles inequalities can b e separated in p olynomial time

Pro of

The numb er of p otential links L is p olynomial of order O jV j We set up a link graph that

has the links as its no des this device will in a second turn out to b e a subgraph of G Two

Set Packing Relaxations

links are connected by an edge if and only if they are joined byaK To assign weights to

the links we calculate for eachlinkL the shortest even path P in G with an even number

i i

of no des that connects the two endp oints of the link here shortest means shortest with

resp ect to the length function x x for all edges ij E L P forms a shortest

i j i i

o dd cycle C through L with resp ect to the length x x x

i i j j C

j C

ie a longest o dd cycle C through L with resp ect to the length x We set the weight

i i C

of link L to the value x obtain the link graph as a subgraph of GfC g some edges

i C i

that corresp ond to nonlink conicts are p ossibly missing and detect a violated o dd cycle

inequality in the link graph if and only if a violated cycle of cycles inequality in G exists

eventuallywersthaveto separate the edge inequalities of the link graph

if one of the A cycle of cycles inequality will in general not be facet inducing for example

cycles has a chord that joins two nonlink no des But one can come up with conditions that

ensure this prop erty The most simple case is where the cycles C are holes all no de disjoint

and the only edges that run b etween dierent holes b elong to the links ie wehave a hole

of holes In this case the cycle of cycles inequality is easily shown to b e facet inducing using

standard techniques like noting that every edge in a hole of holes is critical

Prop osition Facet Inducing Cycle of Cycles Inequalities

If every cycle in a cycle of cycles inequality is a hole and the only edges that run between

dierent holes emerge from the links then the cycle of cycles inequality is facet inducing for

the set packing p olytop e P G asso ciated to the supp ort graph G of the inequality

SSP

We want to give now an alternative proof for the faceteness of the hole of holes inequality

The technique that we are going to demonstrate works also for other constructions of this

typ e It is our aim to give an example how aggregation techniques although not designedfor

facetial investigations can sometimes lend themselves to results in this direction

Pro of of Prop osition

The idea of the pro of is to exploit the comp osition structure of a hole of holes C fC C g

To this purp ose it is convenient to consider C sometimes as a hole in the conict graph Gin

which case wewant to denote it by C and sometimes as a structure in the original graph G

and then wewant to use the original notation C The rst step is to lo ok at the inequalityas

a linear form in the image space of the aggregation namely as the o dd cycle inequality

x k

vC C

of the set packing p olytop e P C asso ciated to the odd hole C As constraint is a

SSP

C there are k anely indep endent incidence vectors x r k of facet of P

SSP

set packings on the induced face Likewise each of the individual holes C has a set of jC j

i i

anely indep endent incidence vectors of set packings x in the graph C s jC j that

i i

are tight for the o dd cycle inequality asso ciated to C ie

jC j

x jC j i k s jC j

i i

Moreover there is an incidence vector x of a set packing in C suchthat

jC j

jC j i k x

The Set Packing Problem

Note that

jC j

x jC j i k s jC j

i i

jC j

x jC j i k

is r

We will use the vectors x to expand the vectors x into a set of jC j anely indep endent

incidence vectors of stable sets in C that are tight for the hole of holes inequality in question

To this purp ose we can assume without loss of generalitythat

x r k

ie the r th comp onent of the r th aggregated incidence vector x is one We now blow

r rs C

x into jC j vectors y R s jC j dened as up eachvector

r r

i r

x if x

i r

y i k

x if x and i r

r is

x and i r x if

rs rs C

y indexes the subvector of y R with all comp onents that corresp ond to the hole C

In other words We take each incidence vector x and substitute for each of its comp onents

r r r is i i

x x x i k a vector x If we take x if we take x The only

C C C

i i i

exception to this pro cedure is co ordinate r where we do not only substitute x but try all

r rs rs

x for all s jC j p ossibilities x In all cases however y

jC j vectors y It is easy to see that these expansions of This results in a total of

stable sets by stable sets are incidence vectors of stable sets in C and that they are tight

for the hole of holes inequality under consideration We claim that they are also anely

indep endent For supp ose not then there are multipliers not all zero that sum up to

zero such that y But this implies that

jC j

X X X X

rs r r

y x x

rs rs rs

rs rs

r s

and ane indep endence of the aggregated vector x yields

jC j

r k

Considering the rows of y that corresp ond to the individual holes C we obtain

rs r

y r k

is i

As for i r the vectors y x are constant for all s these equations simplify to

jC j jC j jC j

r r

X X X X X

rs rs is i

y x r k y x

rs rs is is

C C

r r

s s ik s

i r

see

and imply a contradiction Thus the incidence vectors y were indeed anely

indep endent and the hole of holes inequality facet inducing for its supp ort

Set Packing Relaxations

Chain Inequalities

We have seen in the preceding subsections a variety of derivations of classes of inequalities

from cycle and clique inequalities of appropriate set packing relaxations Togive an example

of a dierent combinatorial typ e weshow in this subsection that a family of chain inequalities

that were intro duced by Tesch can be seen as strengthened see page expansions

of Nemhauser Trotter s antiweb inequalities see also Laurent

A k chain C is similar to a chorded cycle with k no des k the dierence

is that the twochords k and k are replaced with the single edge k see

Figure An antiweb C k t is a graph on k no des k suchthatany t successive

no des i i i t form a clique see Figure Chains are very similar to chorded

cycles these in turn coincide with the class of antiwebs C k

Chains and antiwebs give rise to inequalities for the set packing p olytop e The chain and

antiweb inequalities state that

X X

k k

x x

i i

iC kt

Figure A Chain

Figure The Antiweb C

Theorem Chain Inequalities

Let C be k chain P the corresp onding set packing p olytop e G V E the rank

SSP

conict graph and P G the rank set packing relaxation of P Then

SSP SSP

Every chain inequality for P is the expansion of a strengthened antiweb inequality for

SSP

P G

SSP

Pro of

Consider in G the k no des

v Gf k g v Gfig i k and v Gfk g

and let W fv v g The reader veries that W induces an antiweb in G more

precisely

GWC k

The Set Packing Problem

An expansion of the antiweb inequality corresp onding to GWyields

x x x x x

k k

A nal strengthening of this inequality reducing the righthand side by one see page

yields the desired chain inequality The validity of the strengthening can be inferred in a

similar way as in the pro of of Theorem

Some Comp osition Pro cedures

While the examples of the preceding subsections had analytic avour we study in this sub

section applications of set packing relaxations to constructive approaches to the stable set

p olytop e Our result is that certain comp osition pro cedures of the literature have a natural

interpretation in terms of set packing relaxations

The general principle b ehind composition appr oaches is the following One considers some

graph theoretic operation to construct a complex graph G from one or more simpler ones

and investigates the p olyhedral consequences of this op eration Such consequences can be

i to obtain analogous op erations to construct valid or facet dening inequalities for G from

known ones for the original graphs or in rare cases ii to obtain a complete description

of P G from likewise complete descriptions of the antidominants of the set packing

SSP

p olytop es asso ciated to the original graphs A survey on comp osition metho ds for the set

packing problem can b e found in Section of this thesis

Op erations of typ e i that give rise to facets are called facet producing procedures and we

study three examples of this typ e in the remainder of this subsection weinvestigate only their

validity The graph theoretic comp osition technique b ehind all of them is no de substitution

G replacing one or several no des by graphs and the in dierentvariants Given is some graph

aected edges by appropriate sets of edges one obtains a new graph G The facet pro ducing

b b

pro cedure asso ciated to such a substitution translates validfacet dening inequalities for

P Ginto validfacet dening inequalities for P G

SSP SSP

This concept has an obvious relation to expansion Namely consider the expansion

T T

a x a x

of an inequality for the rank relaxation P G of some graph G One obtains the supp ort

SSP

T T

a of the expansion from the supp ort graph Gsupp a of the aggregated graph Gsupp

inequalitybya sequence of no de substitutions and identications Constructing inequalities

in this way means thus to lo ok at a given graph G as the conict graph or a subgraph of it

G of some graph G that can b e constructed from

construct G such that G GG

This technique to start with the conict graph and construct a suitable original graph is

our interpretation of comp osition in terms of aggregation

Set Packing Relaxations

Figure Applying a Comp osition Pro cedure

We turn now to the examples and start with a pro cedure of Wolsey Given a graph

G VE with no des V f ng the op eration constructs a new graph G V E

G by replacing no de n with a path n nn involving two new no des n and from

n such that n is adjacent to some subset of neighb ors of the old no de n while

n is adjacent to the remaining neighb ors Figure shows an example where no de of

a graph is replaced by the path and the new no de is connected to the old neighbors

f g of The pro cedure asserts that if a x was a valid inequalityfor P G

SSP

the constraint

a x a x a x a

n n n n n

holds for P G This inequality can b e obtained from a rank relaxation of G that involves

SSP

V V

the aggregation scheme R R dened as

x i n

x x x i n

n n n

maps eachnodeonto itself except for the path n n n which is aggregated into a

single no de One easily checks

G GG Lemma Comp osition Pro cedure I

a x yields inequality We remark that this argument do es not An expansion of

show that this pro cedure translates facets into facets

Our second example is due to Wolsey and Padb erg The pro cedure joins an

additional no de n to all no des n of the given graph G VE and the graph

G V E arises from this join by sub dividing each of the new edges n i with a no de

n i In this case the inequality a x a x for P G gives rise to the constraint

i i SSP

n n n

X X X

a x x a x a

i i ni i n i

i i i

P G and is in fact even facet inducing if a x was Figure shows an for

SSP

application of this technique to the graph G on the left side with no des adding the

grey no de in the middle results in a certain graph G that will b e explained in a second

The Set Packing Problem

GG G

Figure Another Comp osition Pro cedure

V V

To obtain inequality from a rank relaxation we consider the scheme R R

dened as

x x x i n

i ni n

x i n

where wehaveset G V E V fn g E

Lemma Comp osition Pro cedure II G GG

In other words the conict graph G of G coincides with G augmented by an additional no de

P n

b b

n that is not connected to any other no de Obviouslyany inequality a x that

i i

is valid for P Gisalsovalid for P G

SSP SSP

It is now not true that an expansion of the inequality a x for P G yields the

i i SSP

desired inequality but we get it with one additional strengthening typ e argument

a x is valid for P G the stronger inequality This argument is that if

i i SSP

a x x

i i n

b b

is p erhaps no longer valid for P G but it is valid for P G An expansion of this

SSP SSP

inequality yields the inequality of interest but again no facetial result

As an example of a much more general comp osition technique we consider now the substi

G by some graph G such that the resulting graph G V E is the tution of a no de v of

union of G v and G with all no des of G joined to all neighbors of v in G Substitutions

v v T

of this typ e were studied by Chvatal who showed not only that if a x is a facet

of P Gandb x a facet of P G the inequality

SSP SSP v

X X

a b x a x

v u u u u

uV G

u v V

u v

b b

P G but that all facets of P G are of this form Note that this op eration is valid for

SSP SSP

subsumes the famous multiplication of a no de to a clique of Fulkerson and Lovasz

that plays an imp ortant role in studying the p olyhedra asso ciated to perfect graphs

Set Packing Relaxations

To derive the validity of inequalities for xed but arbitrary v V and b x as ab ove

V V

from a set packing relaxation we consider the aggregation scheme R R given as

x u v

b x

w w

w V G

b x

u v

is b ounded by one in every comp onent integral in all co ordinates dierentfrom v but not

integral in v and in particular not a rank aggregation is our only nonrank example in this T

b x is a support of P G ie there is an incidence vector x of a section But if

SSP v

stable set in G such that the inequality b x is tight the aggregate P G has not

v SSP

only integer and thus zeroone vertices only but in fact

b b

Lemma Comp osition Pro cedure III P G P G

SSP SSP

a x yields Chvatals Once this relation is established an expansion of the inequality

inequality but not an equivalent result ab out complete descriptions

Pro of of Lemma

We prove rst that P G is integral The pro of is by contradiction ie supp ose

SSP

P G is not integral Then there must be a non integer vertex x where x is a

SSP

P G Note that x and so must b e x The only fractional comp onent vertex of

SSP

of x canbe x and it must b e nonnull By assumption there exist incidence vectors

y and y of stable sets in G such that

T T

b y and b y

with The vectors x and x that arise from x by replacing x y and y are again incidence

vectors of stable sets in G b ecause x is nonnull and the stable set supp x has a no de in

G But then

T T

x b x x b x x

b b

is not a vertex a contradiction

P G P Gistonotethat x x holds for The last step to establish

SSP SSP u w

anyvertex x f g of P G if and only if uw E G

SSP

The Set Covering Problem

We prop ose in this section a set packing relaxation of the set covering problem that gives

rise to p olynomially separable classes of inequalities This is imp ortant for two reasons

i Set covering deals with general indep endence systems see Section of this thesis while

many problems in combinatorial optimization arise from sp ecial indep endence systems hence

the set covering results carry over Unfortunatelyhowever ii very few classes of p olynomial

time separable inequalities for the set covering problem are known we are only aware of the

o dd hole inequalities see Subsection Nobili Sassano s k projection cuts Balas

g ChvatalGomory cuts see Caprara s conditional cuts and certain classes of f

Fischetti see also Schulz Section for some classes of this t yp e

The Set Covering Problem

The need to develop cutting planes for the set covering p olytop e was the starting p oint for our

work on set packing relaxations Wehave implemented one version of such a pro cedure for use

in a branchandcut algorithm for set partitioning problems details ab out and computational

exp erience with this routine are rep orted in Chapter of this thesis

The set covering problem SCP is the integer program

T n

SCP min w x Ax x Z

mn n

where A f g and w Z The asso ciated p olyhedron is denoted in this section by

P or P A For a survey on set covering see Chapter

SCP SCP

The set packing relaxation for SCP that we suggest is based on an exp onential conict graph

G V E that records pairwise conicts of substructures of the matrix A We take the set

fng S

V where denotes the powerset of some set S of all subsets of column indices

n V

of A as the no des of G and dene an aggregation scheme R R as

x J f ng x

j J

We drawanedge between two not necessarily disjoint sets I and J of columns when their

union coversarowofA or equivalentlysomevariable in I J has to b e set to one

x x P Z IJ E i I J A x

J SCP i I

Lemma Set Packing Relaxation of the SCP P P G

SCP SSP

This set packing relaxation has b een considered by Sekiguchi in a sp ecial case He

studies matrices A with the prop erty that there is a partition W of the column indices

v f ng into nonempty column sets v such that the supp ort of eachrow A is

the union of exactly twosuch column sets ie A u v W u v supp A u v

r r

Figure shows an example of a matrix that has sucha Sekiguchi partition

B C

W f g fg f g fg

B C

fg f g

Figure A Sekiguchi Partitionable Matrix

Using essentially the same technique as we did to prove Prop osition ab out the faceteness

of hole of holes inequalities Sekiguchi shows that for a matrix A that has a

Sekiguchi partition W it is not only true that

P P GW

SCP SSP

P GW but even more that the facets of P are exactly the expansions of the facets of

SSP SCP

We remark that Sekiguchi considers in his pro of the as we would say aggregation scheme

n W

R R dened as

x x v W

v i

that is complementary to ours in the sense that

Set Packing Relaxations

We mention the odd hole inequalities for the SCPsee eg Cornuejols Sassano as

one example for a class of inequalities that can b e obtained from a set packing relaxation in

the sense of Sekiguchi

In this context of set covering the term odd hole is commonly used to refer to the edgeno de

k k

incidence matrix Ak AC k R of the circulant C k

if j i or j i mo d k

else

The asso ciated odd hole inequality asserts that

x k

is valid for P Ak

SCP

Prop osition

Let Ak be an odd hole P the corresp onding set covering p olyhedron G the

SCP

conict graph asso ciated to Ak and P GW the Sekiguchi relaxation of P

SSP SCP

where W ffigji k g is the unique Sekiguchi partition of Ak Then

The odd hole inequality asso ciated to P is the expansion of an odd cycle inequality for

SCP

P GW

SSP

We omit the simple pro of of this prop osition and pro ceed with an example of an expanded

cycle inequality that can not b e obtained from a Sekiguchi relaxation we call this larger class

of inequalities from expansions of cycle inequalities for G aggregated cycle inequalities

f g

f g fg

B C

f g f g

Figure A Not Sekiguchi Partitionable Matrix and an Aggregated Cycle

The matrix A on the left of Figure gives rise to a cycle C in G formed by the no des v

f g v fg v f g v f gandv f g A is not Sekiguchi partitionable

b ecause row A calls for the sets supp A n supp A f g and supp A supp A fg

as elements of the partition but these sets are not disjoint An expansion of the odd cycle

inequality corresp onding to C yields

x x x x x x x x x x

x x x x x x x x x

The Multiple KnapsackProblem

Turning back to general case and lo oking at the separation of inequalities for P from the

SCP

P G we can obtain p olynomially separable classes by restricting set packing relaxation

SSP

attention to no de induced subgraphs GWof the conict graph of p olynomial size A heuristic

way to do this is to split the supp ort of eachrow A into two equal sized halves

supp A I I

with resp ect to a given fractional covering x ie we split such that A x A x and take

iI I

W as the set of these halves

I j i mg W fI

The idea b ehind this pro cedure is to i obtain a reasonable p olynomial number of m

no des in fact in our computations a lot of these always turned out to b e identical with values

of x x close to that lead ii with some probability not only to a signicantnumber

of edges at all but hop efully even to tight edges of the set packing relaxation which

in turn iii oers some p otential to identify violated inequalities We have implemented

this pro cedure to separate aggregated cycle inequalities in a branchandcut co de for set

partitioning problems for more implementation details and computational experience with

this routine see Chapter of this thesis

Another separation idea that suggests itself would b e to derive inequalities from submatrices

of A But in contrast to the set packing case such inequalities are in general only valid for

their row andor column supp ort They havetobelifted to b ecome globally valid and wedo

not knowhow to derive eciently separable classes of inequalities in this way

We close this subsection with an attempt to demonstrate the exibility of the concept of the

set packing relaxation P G by stating a result of Balas Ho on cutting planes

SSP

from conditional bounds in set packing relaxation terminology

Balas Ho assume that some upp er bound z on the optimum ob jective value of the set

covering program SCP is known In this situation they consider some family of variable

index sets W V and investigate conditions that ensure that at least one of the corresp onding

aggregated variables x v W has a value of one for any solution x with a better objective

value than z If this condition can b e established Balas Ho suggest argumen ts and

algorithms based on LP duality the conditional cut

supp A nv i

r v

can b e used as a cutting plane Here for each column set v A is an arbitrary rowofA

r v

Note that conditional cuts are again of set covering typ e

The Multiple Knapsack Problem

In this section we investigate a set packing relaxation of the multiple knapsack problem in

an exp onential space It will turn out that the validity of certain classes of extended cover

item and combined cover inequalities can b e explained in terms of a single conict of two

knapsack congurations As references to the multiple knapsack problem we give Wolsey

the textb o ok Martello Toth Chapter Ferreira Martin Weismantel

see also the thesis of Ferreira and the survey article Aardal Weismantel

Set Packing Relaxations

The multiple knapsack problem MKP is the integer program

X X

w x MKP max

iI k K

i a x k K

x i I ii

k K

I K

ii x f g

Here I f ng is a set of items of nonnegative integer weights and prots a w Z

that can b e stored in a set K of knapsacks of capacity each Asso ciated with the MKP is

the multiple knapsack p olytop e P

MKP

The set packing relaxation that we prop ose involves the following conict graph G V E

I K S

G has the set V where denotes the powerset of some set S of all sets of p ossible

itemknapsack pairs as its no des We will call such a set of itemknapsack pairs an item

knapsack conguration To dene the edges of the conict graph we consider the aggregation

I K V

scheme R R dened as

jI vj x x

ik v

Here I vfi I k K ik vg denotes the set of items that app ear somewhere in the

conguration v x is one for some solution x of MKP if and only if x assigns al l items

in v to feasible knapsacks with resp ect to v ie all items i I v of the conguration satisfy

x for some i k v Two congurations u and v are in conict and we draw an edge

I K

between them if x x holds for all x P Z ie it is not p ossible to

u v MKP

simultaneously assign all items in u and v to feasible knapsacks

Lemma Set Packing Relaxation of the MKP P P G

MKP SSP

We shownow that the classes of extended cover inequalities and combined cover inequalities

of Ferreira Martin Weismantel arise from expansions of edge inequalities of this set

packing relaxation our discussion refers to Ferreira s description of these inequalities

An extended cover inequality involves two congurations v and v of the form

v I fk k g and v I fk g

with two knapsacks k and k and two sets of items I and I In this situation it is in general

not true that x x holds but one can lo ok for combinatorial conditions that

v v

ensure this inequality Ferreira page assumes that

i I forms a cover for knapsack k ie the items in I do not all t into k and that

ii I fig forms a cover for knapsack k for each item i I

Actually he assumes also I I i means that if x ie all items I of

the knapsacks k and k then at least one item of I the rst conguration are assigned to

must b e assigned to k and then x due to ii This implies the validity of the edge

inequality x x that expands into the extended cover inequality

v v

X X X

jI j jI j x x x

ik ik ik

iI iI iI

The Programming Problem with Nonnegative Data

Of similar avour are the combined cover inequalities This time the congurations are

v I fk g I fk g I fk g and v I fk g

with three dierent knapsacks k k andk and satisfying confer Ferreira page

i I I I and jI I j

ii I is a cover for k I acover for k and

iii I fig is a cover for knapsack k for eachitemi I

x and i together imply that at least Ferreira assumes again I I

x as for one item from I must b e assigned to knapsack k and then ii results in

the extended cover inequalities Expanding the edge inequality x x we obtain

v v

the combined cover inequality

X X X X

x x x jI j jI j x

ik ik ik ik

iI iI

The following theorem summarizes our results on extended and combined cover inequalities

Theorem Extended and Combined Cover Inequalities

Let MKP b e a multiple knapsack problem P the asso ciated multiple knapsack p olytop e

MKP

and P G the set packing relaxation of P

SSP MKP

i Every extended cover inequality for P is the expansion of an edge inequality for

MKP

P G SSP

ii Every combined cover inequality for P is the expansion of an edge inequality for

MKP

P G

SSP

The Programming Problem with Nonnegative Data

We have seen in the previous sections examples of set packing relaxations for sp ecial combi

natorial optimization problems To give a perspective in a more general direction we want

to draw the readers attention nowto a set packing relaxation for a class of integer pro

gramming problems that was suggested by Bixby Lee This construction assumes

only nonnegativity of the constraint matrix but no particular structure it yields clique o dd

cycle etc inequalities in the natural variables

Set packing constraints of this typ e form one of the rare families of structural cuts for general

integer programming problems ie cuts that are derived by searching detecting and utiliz

ing some combinatorial structure in an a priori unstructured constraint system Set packing

relaxations try to set up a conict graph the famous single knapsack relaxation of Crowder

Johnson Padb erg analyzes the diophan tine structure of an individual row Padb erg

van RoyWolsey s owcovers are based on combinatorial prop erties of xed charge

problems and the feasible set inequalities of Martin Weismantel come from inter

sections of several knapsacks These classes of structural cuts are the rst of only three typ es

of to ols to solveinteger programs by branchandcut Enumeration is unfortunately the

second and the third consists of general cutting planes for integer programs Gomory

s cuts see Balas Ceria Cornuejols Natra j for an exciting recen t renaissance

liftandpro ject cuts see Balas Ceria Cornuejols or Lovasz Schrijver s

matrix cuts To put it brief There is signicant interest in identifying further families of

structural cuts for general integer programs

Set Packing Relaxations

One class of integer programs with an emb edded set packing structure consists of programs

with nonnegative constraint systems

T n

IP max w x A x b x f g

Here A Z and b Z are a nonnegativeintegral matrix and righthand side vector

and w Z is an integer ob jective no further structural prop erties are assumed The p olytop e

asso ciated to this program is denoted by P

Bixby Lee prop ose for suc h programs the following natural set packing relaxation

The conict graph G V E of the relaxation has the column indices of the matrix A

ng The edges are dened in terms or if wewant the variables as its no des ie V f

n n

of the identity aggregation scheme R R that has

xx i n

i i

There is an edge b etween two columns i and j if and only if not b oth of them can b e contained

in a solution to IP at the same time ie

ij E A A b x x x P Z

i j

Lemma Set Packing Relaxation of IP Bixby Lee P P G

SSP

The natural set packing relaxation yields clique o dd cycle and other set packing inequalities

in the original variables An extension of the natural set packing relaxation to the mixed

integer case is currently studied byAtamturk Nemhauser Savelsb ergh

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Index of Part

antiblo cking pair Symb ols

ofmatrices g ChvatalGomory cut f

ofpolyhedra program

antiblo ckingtheory matrix

companion TDI system chord in a cycle

companion theorems chorded cycle in a graph

maxmax inequality chorded cycle inequality

minmax equality for the clique partitioning p olytop e

separation perfectgraphtheorem

colorable hypergraph strong minmax equality

strong p erfect graph conjecture

antidominant

acyclicarc set

of the set packing p olytop e

acyclic sub digraph p olytop e

antichainina poset

fence inequality

antihole

separation

generalized

Mobius ladder inequality

ina graph

separation

N index

o dd cycle of dipaths inequality

antihole inequality

o dd cycle of diwalks inequality

faceteness

separation

for the set packing polytope

acyclic sub digraph problem

N index

set packing relaxation

separation

adjacency of vertices

antiholeperfectgraph

on the set packingpolytope

antiweb

aggregate of a p olytop e

generalized

aggregated cycle inequality

ina graph

for the set covering p olytop e

antiweb inequality

separation

faceteness

aggregation scheme

for the set packing p olytop e

almost ideal matrix

approximation algorithm

almost integral p olyhedron

for the set packing problem

almost p erfect

matrix B

graph balanced

amalgamation of two graphs matrix

antiblo cker hypergraph

of a matrix matrix

of a polytope recognition

INDEX OF PART

circulant in a graph bipartitegraph

evenk covering problem

o ddk

matching problem

circulant inequality

bipartite matching problem

forthe maxcut polytope

bipartite no de covering problem

separation

blo cker

circulant matrix

of a matrix

claw free graph

of a p olyhedron

rank facets of the set packing p olytop e

blo cking pair

of a claw free graph

of matrices

claw ina graph

of polyhedra

clique

blo cking theory

generalized

ideal matrix conjecture

ina graph

max owmin cut equality

clique covering numb er of a graph

minmin inequality

clique identicationingraphs

strong maxmin equality

clique inequality

widthlength inequality

faceteness

blossom inequality

for the set packing p olytop e

faceteness

N index

for the set packing p olytop e

separation

of a linegraph

clique numb er of a graph

separation

clique partition

blossom p erfect graph

clique partitioning p olytop e

b ound for an optimization problem

chorded cycle inequality

separation

canonical inequality

inequality from an odd cycle of lower

faceteness

triangle inequalities

for the set packing p olytop e

separation

certicate of optimality

clique partitioning problem

chain

set packing relaxation

ina graph

clique p erfect graph

ina poset

closed diwalk in a digraph

chain inequality

closed set in an indep endence system

faceteness

co edge identication in graphs

for the set packing p olytop e

co edge in a graph

p erfect graph

coloring

plup erfect graph

edge coloring theorem of Konig

p erfect graph

edges of a bipartite graph

plup erfect graph

no des of a h yp ergraph

chordina cycle

coloring numberofa graph

chromatic number

column intersection graph

of a graphsee coloring number

of a set packing problem

of a hyp ergraph

combined cover inequality

ChvatalGomory cut

for the multiple knapsack p olytop e

f g ChvatalGomory cut

companion TDI system

in antiblo ckingtheory circuit of an indep endence system

INDEX OF PART

cycle of cliques inequality companion theorem

complement of a graph

for the set packing polytope

complete bipartite comp osition

cycle of cycles inequality

of two matrices

faceteness

complete bipartite graph

for the set packing p olytop e

complete description of a p olyhedron

separation

complete digraph

cycle of dipaths see o dd cycle of dipaths

complete graph

cycle of diwalks see o dd cycle of diwalks

N index

cycle of lower triangle inequalities see odd

comp osition approaches to set packing

cycle of lower triangle inequalities

comp osition of circulants in a graph

cycle of paths inequality

comp osition of circulants inequality

for the set packing polytope

faceteness

separation

for the set packing p olytop e

cycle of upp er triangle inequalities see odd

comp osition of two graphs

cycle of upp er triangle inequalities

comp osition pro cedure

cycle p erfect graph see tp erfect graph

for the set packing p olytop e

conditional b ound

decompositiontree

for the set covering problem

degenerate pro jective plane

conditional cut

deletion of a co ordinate

for the set covering p olytop e

Dilworthstheorem

separation

distance claw free graph

conguration of items and knapsacks

diwalk in a digraph

in a multiple knapsack problem

closed conict graph

of a set packing problem

down monotonicity

of a set packing relaxation

of the set packing p olytop e

contraction of a co ordinate

contraction ofanoddpath

edge coloring

core of a matrix

in a bipartite graph see edge coloring

cover

theorem

for a knapsack

edge coloring theorem of Konig

in a bipartite graph see

edge inequality

KonigEgervary theorem

faceteness

in a hypergraph

for the set packing p olytop e

cover inequality

edge perfectgraph

for the knapsack problem

edge relaxation

covering problem in a bipartite graph

of the set packing problem

critical cut in a graph

edge sub division in a graph

criticaledge ina graph

edgeno de incidence matrix of a graph

critical graph

equivalence

for the set covering problem

of optimization and separation

for the set packing problem

evenk circulant in a graph

cycle

evenk circulant inequality

generalized

forthe maxcut polytope

ina graph see o dd cycle

in an indep endence system separation

INDEX OF PART

generalized antiw expansion of an inequality eb

generalized antiweb inequality

extended cover inequality

faceteness

for the multiple knapsack p olytop e

for the set covering p olytop e

extended description of a p olytop e

generalizedclique

extension of a graph

generalized clique inequality

faceteness

facet dening graph

for the set covering p olytop e

for the set packing p olytop e

generalizedcycle

facet dening inequality

generalized cycle inequality

facet dening matrix

faceteness

for the set covering p olytop e

facet pro ducing graph

generalized set covering problem

for the set packing p olytop e

generalized set packingproblem

facet pro ducing pro cedure

generalized web

for the set packing problem

generalized web inequality

Fano plane

faceteness

feasible set inequality

for the set covering p olytop e

for an integer program

geometria situs graph theory

fence in a digraph

Gomory cut

fence inequality

for an integer program

for the acyclic sub digraph p olytop e

graph

incidencematrix

separation

with b ounded N index k

with b ounded N index k lteroracle

graph theoretic approach

owcover inequality

to the set packing problem

for an integer program

graph theoretic op erations

forbidden minor

amalgamation

fractional

clique identication

covering polyhedron

co edge identication

packingpolytope

comp osition of two graphs

set covering polytope

contraction of an o dd path

set packingcone

edge sub division

set packing matrix cone

extensionofa graph

set packing p olytop e

intersection of two graphs

set partitioning p olytop e

join

fractional covering problem

lexicographic pro duct of two graphs

fractional packing problem

multiplication

fundamental theorem of linear prog

no de substitution

p olyhedral consequences

general cutting plane

replication

for an integer program

subdivisionofa star

generalized antihole

substitution

generalized antihole inequality

sum of two graphs

faceteness

union of two graphs

graphtheory for the set covering p olytop e

INDEX OF PART

liftandpro ject cut H

matrix inequality hp erfect graph

structural cut

half integral vertex of a p olyhedron

intersection graph

Helly prop erty

of a set packing problem

heuristic lifting

intersection of two graphs

Hirsch conjecture for p olytop es

item in a multiple knapsack problem

homogenization

itemknapsack conguration

of the set packing p olytop e

in a multiple knapsack problem

homogenized unit cub e

hubof a wheel

Hungariantree ina graph

join of graphs

hypergraph

colorable

balanced

k fence see fence

hyp omatchable graph

k multicut see multicut

k pro jection inequality

for the set covering p olytop e

idealmatrix

separation

ideal matrix conjecture

K inequality

imp erfection

faceteness

of a matrix

for the set packing p olytop e

of a graph

Konig

improvement direction

edge coloring theorem

in a lo cal search algorithm

KonigEgervarytheorem

incidence matrix of a graph

marriagetheorem

indep endence system

knapsack

relation to set covering

in a multiple knapsack problem

indep endence system p olytop e see set

knapsackcover

covering p olytop e

knapsack p olytop e

inequality

cover inequality

with b ounded N index k

knapsack problem

with b ounded N index k

inequality from an o dd cycle of lower trian

gle inequalities

Lp erfect graph

for the clique partitioning p olytop e

lexicographic pro duct of two graphs

separation

liftandpro ject cut

inequality from an odd cycle of upp er tri

for an integer program

angle inequalities

lifting

forthe maxcut polytope

co ecient

separation

for the set covering p olytop e

integer program

for the set packing polytope

integer aggregation scheme

heuristic

integer program

in pseudo p olynomial time

feasible set inequality

problem

owco ver inequality

sequence

general cutting plane

sequential metho d

Gomory cut simultaneous metho d

INDEX OF PART

minmax equality line graph of a graph

minmax theorem linear ordering p olytop e

minmin inequality linear ordering problem

minimally imp erfect link graph for a cycle of cycles

matrix link in a cycle of cycles

graph lo cal optimum in a lo cal search

N index lo cal search

minimally nonideal improvement direction

matrix lo cal optimum

minor search graph

ofa balancedmatrix

of a graph

marriage theorem of Konig

ofa matrix

matching

of a p olytop e

in a bipartite graph see

Mobius ladder

KonigEgervary theorem

in a digraph

ina graph

with arc rep etition

matching problem

Mobius ladder inequality

in a bipartite graph

for the acyclic sub digraph p olytop e

matrix cone

matrixcut see matrix inequality

separation

matrix inequality

multicut in a graph

for a integerprogram

multicut p olytop e

for the set packing p olytop e

multicut problem

separation

multiple knapsack polytope

with N index k

combined cover inequality

with N index k

extended cover inequality

matroid

multiple knapsack problem

rank facet

item

max cut p olytop e

itemknapsack conguration

circulant inequality

set packing relaxation

separation

multiplication of a no de in a graph

evenk circulant inequality

N separation

N index inequality from an odd cycle of upp er

of a graph triangle inequalities

of an inequality separation

N index oddk circulant inequality

of a graph separation

of an inequality upp er triangle inequality

no de coloring of a hyp ergraph max cut problem

no de covering problem set packing relaxation

ina bipartitegraph max owmin cut equality

no de separator in a graph max owmin cut theorem

no de substitution in a graph maxmax inequality

nonseparable set maxmin equality

inanindependencesystem Meynielgraph

INDEX OF PART

packingproblem O

paleina fence odd K

partiallyorderedset odd antihole see antihole

oddcut ina graph partition in a hypergraph

o dd cycle partitionable graph

ina graph partner of a clique or stable set

chord in a minimally imp erfect graph

N index p erfect graph

o dd cycle inequality antiholeperfect

blossom p erfect graph

faceteness

for the set packing p olytop e perfect

N index perfect

separation cliqueperfect

o dd cycle of dipaths inequality cycle perfect see tp erfect

edge perfect

for the acyclic sub digraph p olytop e

o dd cycle of diwalks inequality hperfect

for the acyclic sub digraph p olytop e Lperfect

separation pluperfect

o dd cycle of lower triangle inequalities tperfect

wheelperfect

for the clique partitioning p olytop e

o dd cycle of upp er triangle inequalities p erfect graph conjecture

for the max cut problem p erfect graph theorem

odd hole p erfect graph theory

circulant matrix perfectmatrix

picket ina fence

ina graph

o dd hole inequality pluperfectgraph

faceteness pluperfect

for the set covering p olytop e pluperfect

separation polarofa set

p olyhedral consequences

o dd wheel in a graph see wheel

o ddk circulant ina graph of graph theoretic op erations

o ddk circulant inequality p olynomial time equivalence

forthe maxcut polytope of optimization and separation

separation p olytop e

extended description

op en ear decomp osition of a graph

optimization poset

p olynomial time equivalence with sep powerset ofa set

aration

primal approach

oracle to the set packing problem

orthogonality inequality pro jective plane

for the set packing p olytop e prop er matrix

N index prop erty of a matrix

separation

prop erty of a matrix

orthonormal representation of a graph

quadratic relaxation P

of the set packing problem packing in a hyp ergraph

INDEX OF PART

of evenk circulant inequalities R

of fence inequalities rank

of inequalities from o dd cycles of lower

in an indep endence system

triangle inequalities

of a graph

of inequalities from o dd cycles of upp er

rank inequality

triangle inequalities

faceteness

of k pro jection inequalities

for the set covering p olytop e

of matrix inequalities

faceteness

of Mobius ladder inequalities

separation

of o dd cycle inequalities

for the set packing p olytop e

of o dd cycle of diwalks inequalities

faceteness

of o dd hole inequalities

for the set packing p olytop e of a claw

of o ddk circulant inequalities

free graph

of orthogonality inequalities

rank relaxation

ofrankinequalities

of the set packing problem

of wheel inequalities

recognition

of wheel inequalities of type I

of balanced matrices

of wheel inequalities of type II

of ideal matrices

p olynomial time equivalence with op

of p erfect graphs

timization

of p erfect matrices

separation oracle

relaxation

separator in a graph see no de separator

set packing relaxation of a combinato

sequentialliftingmethod

rial program

seriesparallelgraph

single knapsack relaxation of an integer

set covering p olytop e

program

aggregated cycle inequality

replicationlemma

separation

replication of a no de in a graph

conditional cut

rimofa wheel

separation

facet dening matrix

schemesee aggregation scheme

generalized antihole inequality

search graph for a lo cal search

faceteness

Sekiguchi partition of a matrix

generalized antiweb inequality

Sekiguchi relaxation

faceteness

of a set covering problem

generalized clique inequality

semidenite relaxation

faceteness

of the set packing problem

generalized cycle inequality

separation

faceteness

of chorded cycle inequalities

generalized web inequality

of aggregated cycle inequalities

faceteness

of antihole inequalities

k pro jection inequality

of blossominequalities

separation

of circulant inequalities

o dd hole inequality

of clique inequalities

rank inequality

of conditional cuts

faceteness

of cycle of cycles inequalities

separation

up monotonicity of cycle of paths inequalities

INDEX OF PART

o dd cycle inequality set covering problem

faceteness conditional b ound

N index relation to indep endence systems

separation Sekiguchi relaxation

o dd hole inequality set packing relaxation

faceteness set packingcone

separation set packingpolytope

of a claw free graph adjacency of vertices

orthogonality inequality antihole inequality

N index N index

separation separation

p olyhedral consequences antiweb inequality

of graph theoretic op erations faceteness

rank inequality blossom inequality

faceteness faceteness

web inequality canonical inequality

faceteness faceteness

wedge inequality chain inequality

faceteness faceteness

wheel inequality clique inequality

N index faceteness

separation N index

wheel inequalityoftype I separation

separation comp osition of circulants inequality

wheel inequalityoftype II faceteness

separation cycle of cliques inequality

set packingproblem cycle of cycles inequality

approximation faceteness

combinatorial relaxation separation

comp osition approaches cycle of paths inequality

conict graph separation

edge relaxation down monotonicity

intersection graph edge inequality

primal approach faceteness

quadratic relaxation facet deninggraph

rank relaxation facet dening inequality

semideniterelaxation facet pro ducing graph

set packing relaxation facet pro ducing pro cedure

set packing relaxation faceteness

conict graph generalizations of wheel inequalities

construction homogenization

of a integer program with nonneg K inequality

ative data faceteness

of a combinatorial program matrix inequalit y

separation of the acyclic sub digraph problem

with N index k of the clique partitioning problem

with N index k of the max cut problem

INDEX OF PART

of the multiple knapsack problem U

union of two graphs

of the set covering problem

up monotonicity

of the set packing problem

of the set covering p olytop e

set partitioning p olytop e

upp er triangle inequality

set partitioning problem

forthe maxcut polytope

simplexalgorithm

simultaneous lifting

single knapsack relaxation

W free graph

of an integer program

web

skeleton of a polytope

generalized

sp oke ofa wheel

ina graph

stabilitynumb er of a graph

web inequality

stable set in a graph

faceteness

stableset polytope see set packing

for the set packing polytope

p olytop e

wedge in a graph

stable set problem see set packing

wedge inequality

problem

faceteness

strengthening of an inequality

for the set packing polytope

strong maxmin equality

weighted covering problem

strong minmax equality

weighted packingproblem

strong p erfect graph conjecture

wheel in a graph

wheel inequality

structural cut

for the set packing p olytop e

for an integer program

generalizations

sub division of a star

N index

substitution

separation

of a nodeina graph

wheel inequalityoftyp e I

of subgraphs in a graph

for the set packing polytope

sum of two graphs

separation

supp ort graph of an inequality

wheel inequalityoftyp e I I

supp orting inequality of a p olyhedron

for the set packing polytope

symmetric dierence of two sets

separation

wheelperfect graph

widthlength inequality

tperfectgraph

TDI total dual integrality

zeroone program

total dual integrality

zeroonehalf ChvatalGomory cut

totallyunimodularmatrix

tournament in a digraph

transitive packing problem

transversal in a hyp ergraph

triangle inequality

forthe maxcut polytope

twochord ina cycle

t wocolorable hyp ergraph

twoterminal network

INDEX OF PART

Part II

Algorithmic Asp ects

Chapter

An Algorithm for Set Partitioning

Summary We do cumentinthischapter the main features of a branchandcut algorithm

for the solution of set partitioning problems Computational results for a standard test set

from the literature are rep orted

Acknowledgement We thank Rob ert E Bixby for many discussions ab out outpivoting

and for making this metho d available in the CPLEX callable library

Intro duction

This chapter is ab out the design and the implementation of a branchandcut algorithm for

the solution of set partitioning problems Our co de BCWe assume for our exp osition that the

reader is familiar with the features of such a metho d In particular we do neither discuss the

theoretical background of cutting plane algorithms see Grotschel Lovasz Schrijver

nor the basic features and the terminology of branchandcut co des see Nemhauser Wolsey

Padb erg Rinaldi Thienel and Caprara Fischetti In fact our

algorithm BC is to some extent a reimplementation of Homan Padb erg s successful

co de CREW OPTTheowchart of BC coincides with the one of CREW OPT and the same applies

to the primal heuristic pool and searchtree management and even to the data structures

Thus we elucidate only those parts of our implementation where we see some contribution

This applies to the mathematical core of the algorithm Separation and prepro cessing

Our description is intended to give enough information to allow a reimplementation of the

routines in BCWe do however neither discuss software engineering and programming issues

nordowe rep ort the computational tests that guided our design decisions

Rob ert E Bixby Dept of Math Univ of Houston TX USA Email bixbyriceedu

An Algorithm for Set Partitioning

Our rst contribution is a separation routine Wehave implemented a new family of cutting

planes for set partitioning problems The aggregated cycle inequalities of Section Re

call that these inequalities stem from a set packing relaxation of the set covering problem

Together with Nobili Sassano s k pro jection cuts and the odd hole inequalities for

the set covering polytope see page in Subsection of this thesis that have rep ort

edly been implemented by Homan Padb erg these cuts form one of the very few

families of combinatorial inequalities for the set covering p olytop e that have been used in a

branchandcut algorithm

Our second contribution concerns preprocessing We have extended some known techniques

of the literature and explored their p erformance with probabilistic metho ds We have also

investigated the interplay of iterated use of prepro cessing op erations with a dual simplex

algorithm It turns out that much of the p otential of prepro cessing can only b e realized after

certain degeneracy issues haveovercome Wehavedevelop ed a novel pivoting technique that

resolves this problem completely

Pointers to other recent computational work on set partitioning problems are Atamturk

Nemhauser Savelsb ergh Lagrangean relaxation with iterated prepro cessing W edelin

Lagrangean relaxation with a p erturbation tec hnique and Chu Beasley pre

pro cessing and genetic algorithms

This chapter is organized as follows In Section we discuss prepro cessing We give a

list of prepro cessing op erations from the literature and p erform a probabilistic analysis of

their running time The iterated application of such techniques in a simplex based branch

andcut algorithm runs into an unexp ected obstacle Degeneracy problems prevent us from

removing large redundant parts of the problem without destroying a dual feasible basis We

show how to overcome this problem Separation pro cedures are treated in Section We

discuss implementation details of our routines for the detection of violated clique cycle and

aggregated cycle inequalities Computational results are presented in Section

The subsequent sections resort to the following notation We consider set partitioning prob

lems of the form

SPP min w w x

x f g

mn n

where A f g and w Z areaninteger matrix and a nonnegativeinteger ob jective

function resp ectively is the density of the matrix A is supp osed to be the maximum

numb er of nonzero entries in a column and is the average number of entries in a column

G GA is the column intersection graph asso ciated to A this graph gives rise to termi

nology like the neighb ors j of a column A etc The real number w the oset is a

p ositiveconstant that plays a role in prepro cessing We denote by x an arbitrary but xed

optimal basic solution of the LP relaxation of SPP its ob jective value by z the reduced

w andby F the set of fractional variables of x costs by

Asso ciated to SSP are the set packing and set covering relaxations

T T

SSP max w w x SCP min w w x

Ax Ax

n n

x f g x f g

It is well known that all of these problems are NP hard see Garey Johnson Wedo

not discuss further complexity issues here see EmdenWeinert et al for this topic

Prepro cessing

Preprocessing or presolving is the use of automatic simplication techniques for linear and

integer programs The techniques aim at improving a given IP formulation in the sense

that some solution metho d works b etter We are interested here in preprocessing for branch

andcut algorithms These algorithms have LP reoptimizations as their computational b ot

tleneck and their presolvers try to make this step more eective by i reducing the size of

and ii tightening IP formulations and by iii identifying useful substructures Here

tightness is a measure for the qualityofanIPformulation Wesay that IP is a tighter for

mulation than IP iftheinteger solutions of b oth programs are the same but the solution

set of the LP relaxation of IP is contained in that of IP

There are many ways to put iiii into practice Fixing of variables removing redundant

constraints tightening b ounds reduced cost xing probing and constraint classication

are just a few popular examples of reductions as prepro cessing techniques are also called

Surveys on prepro cessing for IPs can be found in Crowder Johnson Padb erg the

textb o ok Nemhauser Wolsey Section I and the references therein Homan

Padb erg and Suhl Szymanski while Brearley Mitra Williams the

lecture notes of Bixby and Andersen Andersen review closely related LP

presolving techniques Most of these metho ds are simple but amazingly eective as the

ab ove publications computational sections do cument

Sp ecial problems oer additional potential for prepro cessing and set partitioning is one of

the best studied classes of integer programs in this resp ect The following subsections sur

vey prepro cessing techniques for set partitioning discuss ecient implementations analyse

exp ected running times and rep ort some computational results

Reductions

We give next a list of reductions for set partitioning problems that subsumes as far as we

know all suggestions of the literature Each technique describ es in principle a transforma

tion and a back transformation of a given set partitioning problem into another one and a

corresp ondence of optimal solutions but as the reductions are quite simple we state them in

a shortcut informal way as in Andersen Andersen These reductions will b e discussed

in more detail in Subsections

P Empty Column j A

If column j is empty one can

i eliminate column j if w

ii eliminate column j and add w to w if w

j j

P Empty Row r A

If row r is empty the problem is infeasible

P Row Singleton rj A e

If row r contains just one column j one can x x to one ie

i eliminate column j and add w to w

ii eliminate al l columns i j that are neighbors of column j and

iii eliminate al l rows s supp A that are cover ed by column j

An Algorithm for Set Partitioning

P P

P Dominated Column i J A A w w i J

i j i j

j J j J

If column i is a combination of other columns j J and w w one can eliminate

i j

j J

the dominated column i

P Duplicate Column i j A A w w i j

i j i j

al and w w one can eliminate column i If two columns i j are identic

i j

P Dominated Row rs A A r s

r s

If row r is contained in row s one can

i eliminate al l columns j suppA A and

s r

ii eliminate the dominated row r

P Duplicate Row rs A A r s

r s

are identical one can eliminate the duplicate row r If two rows r s

P Row Clique rj supp A j j supp A

r r

If al l columns in row r areneighbors of a column j notinrow r one can eliminate column j

T T

P Parallel Column rsij A A e e r s i j

r s

i j

If two rows r s have a common support except for two elements i j one of them contained

in r ow r and the other in row s the variables x x are paral lel and one can

i j

i eliminate columns i and j if they are neighbors ie ij E G

ii merge column i and j into a compound column A A with cost w w otherwise

i j i j

P Symmetric Dierence rstj A A A e r s r t s t

r s t

If row r contains al l columns that are in row s but not in t and some column j that is in

row t but not in s one can eliminate column j

P Symmetric Dierence rst supp A suppA A r s r t s t

r s t

If row r covers the symmetric dierence of rows s and t one can

i eliminate al l columns j suppA A in the symmetric dierence and

s t

ii eliminate row s

A P Column Singleton rj A e j supp A j s A e

s j r r r

If column j is a unit column e and either row r is a doubleton has only two nonzero

zero is covered by some other row sone can elements or row r with a set to

i substitute x a x in the objective to obtain w w w w A x

j ri i j j r

i j

ii eliminate column j and

iii eliminate row r

The next two reductions assume knowledge of an upper bound z

z min w x

x f g

on the optimal ob jective value of the set partitioning problem and knowledge of LP infor

mation P requires an optimal basis of the LP relaxation of SPP and asso ciated data in

particular the ob jectivevalue z the solution x and the reduced costs w P lower b ounds z

on the values of certain LPs For these reasons rules P and P are called LP based

Prepro cessing

P Reduced Cost Fixing j x z z w x z z w

u j u j

j j

i If x and z z w one can eliminate column j

u j

ii If x and z z w one can x x to one

u j j

x z z min c x Ax x x x x f g P Probing j z

j u j j j

i If z z min w x Ax x x one can x x to one

u j j

ii If z z min w x Ax x x one can eliminate column j

u j

Some of these reductions are folklore and do not haveagenuine origin in the literature But

however that may b e PP app ear in Balinski page in a set co vering context

PP andPinGarnkel Nemhauser and PP in Balas Padb erg P

is due to Homan Padb erg substitution techniques lik eP are discussed by Bixby

P was intro duced by Crowder Johnson Padb erg probing techniques like

P are mentioned in Suhl Szymanski some related pro cedures of similar avour in

Beasley for set covering problems

Given all these reductions the next point is to devise a go o d strategy for their application

As the application of one rule can result in additional p ossible simplications for another

rule one usually applies reductions PP in a lo op doing another pass until no further

simplications can be achieved P and P can be applied at any other poin t Once the

LP andor b ounding information is computed these reductions are indep endent of the other

rules

Table gives an impression of the signicance of prepro cessing for the solution of set

partitioning problems The gures in this table were obtained by prepro cessing the Homan

Padb erg acs test set of set partitioning problems from airline crew scheduling

applications with the prepro cessing routines of our co de BC that uses a subset of reductions

PP The rst column in Table gives the name of the problem and the next three

columns its original size in terms of numb ers of rows columns and matrix density ie

the p ercentage of nonzero elements in the matrix Applying some of the non LPbased

prepro cessing rules PP the problems are reduced as indicated in the three succeeding

Presolved columns The remainder of the table go es one step further After solving the

LP relaxation of the prepro cessed problem and calling some primal heuristic the problem is

prepro cessed again This time knowledge of the LP lower b ound z and the upp er b ound z

from the heuristic allows also the use of LPbased techniques in this case reduced xing P

The results of this second round of prepro cessing using a subset of reductions PP are

rep orted in the Presolved LPbased section of the table The success of the LP based

metho ds dep ends on the size of the gap between the heuristic upp er b ound z and the LP

bound z and this gap z z z given as a p ercentage of the upp er b ound the lower

u u

p ossible improvement of z is rep orted in column Gap A value of means that no

valid solution is known Sometimes the LP relaxation is already integral and the problem is

solved In this case indicated by the entry LP in the Gap column it is not necessary to

compute a further upp er b ound or p erform a second round of prepro cessing hence the dashes

in the corresp onding prepro cessing columns One problem nwwas even solved in the rst

prepro cessing phase such that not a single LP had to b e solved this outcome is indicated by

the entry PP in column Gap The nal Time column gives the sum of the prepro cessing

times for dep ending on the problem one or two calls to the prepro cessor in CPU seconds on a Sun Ultra Sparc Mo del E

An Algorithm for Set Partitioning

Original Presolved Gap Presolved LPbased Time

Name

Rows Cols Rows Cols Rows Cols Sec

nw LP

aa LP

us LP

nw LP

us LP

nw LP

nw PP

nw LP

Table Prepro cessing Airline Crew Scheduling Problems

Prepro cessing

The gures in Table show that already without LPbased information the problem size can

often b e reduced substantially by rules PP Using additional LP and heuristic information

leads often but not always to a further reduction in problem size of ab out an order of

magnitude and the prepro cessing can be p erformed in a short time Note that the matrix

densityisessentially unaected by prepro cessing ie it is not true that only very sparse or

dense parts are removed The extent of the reductions can b e explained as a consequence of

the generation of the acs problems The instances are the output of an automatic heuristic

and randomized column generation pro cess that tends to pro duce redundant formulations for

various reasons that we can not discuss here

The second goal of prepro cessing namely tightening of the formulation could however not

be achieved In all cases except nw the optimal ob jectivevalues of the LP relaxations of

the original and the prepro cessed problem are identical The values are not rep orted in the

table And in fact strictly dominated columns for example can not b ecome basic in an

optimal solution anyway and neither do es their identication provide information that is not

also given by the LP solution nor do es elimination of dominated columns help in the sense

that it leads to a dierent LP solution One can check that similar conclusions hold also for

most of the other prepro cessing rules only P P andP can p otentially x variables to

values that would not automatically b e assigned to them by an optimal LP solution

The last two paragraphs argued that the eect of prepro cessing set partitioning problems is

less a tighter LP relaxation than a reduction is problem size There are three main advantages

of solving smaller problems in a branchandcut context First a b etter use of the cache If

large contiguous parts of the problem data can be transferred into highsp eed memory list

pro cessing op erations like the computation of inner pro ducts can b e carried out much more

eciently Note that some care has to be taken to prot from this eect it is in particular

not enough to x or eliminate variables just by adjusting b ounds b ecause this can result in

useless data b eing not only transferred into and out of the cache but also in clogging it

Instead xed columns must b e purged from memory that is accessed for calculations in the

cache A second advantage is that a reduction in the number of rows results in a smaller basis

and this sp eeds up the simplex algorithm Third it is of course also true that elimination

reduces the number of arithmetic op erations Considering problem us for example it is

clear that pricing out columns is much faster than pricing out one million even if all

illustrate these eects we can of the redundant ones are xed by b ound adjustments To

compare the total time to solve the LP relaxations of the original acs problems with the

time needed to solve the presolved instances Prepro cessing halves LP time from

to seconds just as it halves the number of nonzeros and this trend can safely be

extrap olated But simplex iterations as exp ected in the light of the ab ove discussion are

nearly unchanged with in comparison to without prepro cessing using the dual

simplex algorithm of CPLEX steep est edge pricing and turning o the prepro cessing

capabilities of this co de again on a Sun Ultra Sparc Mo del E The numb ers for problem

us are seconds iterations to seconds iterations ie this problem

that makes up ab out half of the test set in terms of nonzeros do es not bias the results of the

ab ove comparison into a misleading direction Aruleofthumb for practical set partitioning

solving is thus that after solving the rst one or two LPs the bulk of the data will have b een

eliminated and the remainder of the computation deals with comparably small problems

We close this intro ductory section with some general remarks on algorithmic aspects Atrst

glance prepro cessing app ears to b e completely trivial b ecause it is so easy to come up with

p olynomial time pro cedures for all rules except P and indeed straightforward implementa

An Algorithm for Set Partitioning

tions work well for small and medium sized problems The only essential issue is to keep an

eye on exploiting the sparsity of the constraint matrix eg by storing columns and rows as

ordered lists of their nonzero entries and this is the only implementation detail mentioned in

most of the literature For large scale problems however naive implementations will not work

anymore For example searching for duplicate columns by comparing all pairs is out of the

question for problems with or more columns although this algorithm has a p olyno

mial complexityof O n op erations assuming that each column has at most some constant

number of nonnull entries Recent algorithms for large scale set partitioning problems of

Homan Padb erg and A tamturk Nemhauser Savelsb ergh th us i use only

simple prepro cessing rules that are ii implemented in a more sophisticated way Both of

these articles contain discussions on design and implementation of prepro cessing routines

tation of the prepro cessing The remainder of this section describ es the design and implemen

mo dule of our set partitioning solver BC Subsection contains some preliminaries on data

structures Subsections to investigate the individual prepro cessing rules PP

We describ e and discuss our particular implementations and do a probabilistic analysis of the

exp ected running times Subsection draws the readers attention to a conict that comes

up in rep eated calls of prepro cessing routines in a branchandcut framework Elimination

of variables andor rows can destroy the dual feasibility of the basis We argue that this

phenomenon is a signicant obstacle and develop a pivoting technique that overcomes the

problem completely The nal Subsection puts the pieces together and describ es the

global layout of the complete prepro cessing mo dule

Data Structures

We will see in the discussions of individual routines in the following subsections that the whole

task of prepro cessing consists of doing various kinds of loops through the columns and rows

of the constraint matrix o ccasionally deleting some of the data The data structures of the

prepro cessing mo dule must allow to p erform these basic op erations eciently and wediscuss

in this subsection some basic issues that come up in this context These explanations are a

preparation for the probabilistic analysis of the following subsection

beg

B C

cnt

B C

ind

Figure Storing Sparse Matrices in Column Ma jor Format

We use a representation of the matrix in row and column major format as ordered and

contiguous lists of the nonzero entries of the columns and rows Figure gives an example

of column ma jor format row ma jor format is obtained by storing the transp osed matrix in

column ma jor format The matrix in the example has rows and columns that are numb ered

starting from Its nonzero entries are stored by row index column wise in ascending

order and contiguously in an array ind The rst three entries and give the row

indices of the nonzero entries of column the next four entries corresp ond to column and

so on the empt y column has of course no entry The arrays cnt and beg are used

Prepro cessing

to lo cate the data for a particular column in the ind array cnti gives the number of

nonzero elements in column iandbegi denotes the starting index for data of this column

in the array ind For more details see eg the manual CPLEX

Column ma jor format allows fast loops through columns As an example consider the fol

lowing Ctyp e pseudo co de to scan column i

int nne

for colpnt colend colpnt f

nne colpnt

some further operations

Note that this lo op requires per nonzero just one comparison of two p ointers that can be

kept in registers one incrementofa pointer in a register and one memory dereference ie

only three operations The slowestofthese is the dereference but this op eration can b enet

from loading the ind array or large contiguous parts of it into the cache Note that this

do esnt work for pointer oriented data structures if data got fragmented in the computers

main memory at least not if no additional precautions are taken Note also that a p ointer

oriented structure requires at least one additional p ointer dereference

The structure also oers various kinds of p ossibilities to eliminate columns conveniently The

simplest metho d is to just set the cnt to zero This technique results in some sup eruous

data in the cnt and beg arrays and chunks of dead data in the ind array with the already

mentioned negative eects At some p oint it hence pays to restore the matrix eliminating

garbage of this typ e this can b e done in time linear in the number of remaining nonzeros

Column ma jor format is of course unsuited for any kind of row oriented op erations To

p erform these eciently as well we store the matrix a second time in row ma jor format We

liketopoint out that this is still more memory ecient than a p ointer oriented representation

b ecause only twoentries are required for each nonzero one ind entry for each nonzero in the

row and one in the column representation

Wehavetopay for the simplicityofrow and column ma jor format when it comes to keeping

the two copies of the matrix synchronized Eliminating columns with any one of the ab ove

mentioned metho ds renders the row representation invalid and vice versa and the only

metho d to make them match again is to transpose the matrix ie to set up the row repre

sentation from scratch This can be done in time linear in the number of nonzeros in two

passes through the matrix The rst pass determines the number of entries p er row and the

second pass puts the elements in their places Tokeep this b o okkeeping eort at a minimum

it is of course advisable to rst p erform all column oriented op erations then transp ose once

do row computations transp ose once and so on This strategy yields reasonable results The

rst round of prepro cessing in Table sp ends seconds out of a total of in

transp osition and these numb ers are also representative for later stages of the computation

This means that wepay a price of ab out in computation time for using the simple row

and column ma jor format It is not so easy to estimate how this compares to other p ossible

data structures b ecause of the eect of additional op erations p erformed in a row or column

scan and wehave not implemented an alternativeversion but we feel that the ab ove consid

erations together with our computational ndings justify the use of row and column ma jor format for prepro cessing purp oses

An Algorithm for Set Partitioning

Probabilistic Analyses

We estimate in this subsection the expected running time of two basic list pro cessing op erations

that we will use frequently in the sequel These results will allow us to compute exp ected

running times for the prepro cessing rules of Subsection Our results are summarized in

the following Table Here eachnumber gives the exp ected running time for the application

of an entire rule ie the value O n log n for rule P gives the exp ected running time for

removing al l duplicate columns and so on Ameansthatwehave not analyzed the rule

Op eration Exp ected Running Time

P EmptyColumns O n

P EmptyRows O m

P Row Singletons O mn amortized

P Dominated Columns

P Duplicate Columns O n log n

P Dominated Rows O m

P Duplicate Rows O m log m

P Row Cliques

M n M M

P Row Clique Heuristic O M n e n

P Parallel Columns

P Symmetric Dierences

P Symmetric Dierences O n

P Column Singletons O n

P Reduced Cost Fixing O n

P Probing

Table Estimating Running Times of Prepro cessing Op erations

The rst list pro cessing op eration that we consider is the lexicographic comparison of two

random sequences of innite length which is supp osed to mo del a test whether two

columns or rows of a random matrix are identical or not We will show that under

certain assumptions this test takes constant exp ected time The second op eration is the

iterative intersection of random sequences of nite length This time we think of a

situation where we w ant to nd common rows in a set of columns or common columns in a

set of rows Again it will turn out that the intersection of random sequences b ecomes

empty fast

Lexicographic Comparison of Two Random Innite Sequences We compute

in this paragraph the exp ected number of op erations for a lexicographic comparison of two

innite random sequences in a certain uniform probabilistic mo del Our analysis will b e

based on the following assumptions

i We lo ok at innite random sequences of zeros and ones where the ones app ear inde

pendently with some probability

stored in a sparse format as ordered lists of the indices of their ii The sequences are

nonnull entries

iii Two sequences from f g are compared lexicographical ly by scanning their index

lists from the b eginning doing as many comparisons as there are common entries in the

two index lists plus one additional comparison to detect the rst dierence

Prepro cessing

Wehave already p ointed out that wewanttouse this setting as a mo del for a lexicographic

comparison of two columns or rows in a matrix of a set partitioning problem In this

context assumptions ii and iii are canonical i assumes identically and indep endently

distributed ones in the sequences Formally such a sequence a b elongs to a probability space

f g AP

that has as its groundset the set of all sequences with an asso ciated algebra A and a

probability distribution P such that the b order distributions are binomial with parameter

ie the two probabilities P a and P a i N exist and have the

i i

stated values This is certainly unrealistic The mo del results in low probabilities for the

existence of duplicate columns and this obviously contradicts the computational ndings of

Table But for want of something b etter we will nevertheless work with i Making

the best of it we can be happy ab out the technical advantage that this mo del has only one

y Our goal will b e parameter the probability which is to b e identied with the matrix densit

to obtain the exp ected numb er of op erations to compare two sequences lexicographically

as a function of Considering innite sequences for this purp ose has the advantage that the

analysis b ecomes indep endent of the number of rows or columns As it takes certainly more

time to compare two innite sequences than two nite ones this results in a mo del that is

valid for lexicographic comparisons of rows and columns

In this not completely sp ecied mo del f g AP consider the following random exper

iment Cho ose two sequences at random and p erform a sparse lexicographic comparison

according to iii Let the random variable Y f g f g N fg denote the

n umb er of comparisons until the rst two indices dier Assumptions iiii suggest that the

probability that such a lexicographic comparison takes k comparisons of individual indices of

nonzeros k should b e

j k

P Y k k N

j k

In this expression is the probability that a common nonzero app ears at a random p osition

in both sequences and is the probability for a common zero There are

p ossibilities to distribute k common ones over the rst j p ositions in b oth sequences

that account for the rst k comparisons and

j k

is the corresp onding probability The nal term is the probability for a dierence

in p osition j that is detected in the k th and last comparison The following theorem assumes

that the mo del f g AP has prop erty

Lemma Lexicographic Comparison of Two Random Sequences

Let and f g AP b e a probability space Let further Y b e a random variable

that counts the number of index comparisons in a lexicographic comparison of two random

elements a b from f g AP If condition holds then

E Y

An Algorithm for Set Partitioning

Pro of The term P Y k can b e simplied to

j k

P Y k

j k

where In this calculation the identity

j k

arises from considering the Taylor series around t of the function

X X

j k j

k j j k

f R t t t t

k k

j k

at t

Since for the function p N k P Y k

is the density of the geometric distribution Geo on with parameter If we consider

motivated by the ab ove arguments the term Geo fk gP Y k as the probability that

exactly k comparisons of indices are necessary to compare two innite sequences that

are stored in sparse format with the ones o ccurring indep endently at each p osition with

probability the exp ected numb er of individual index comparisons is simply the exp ectation

of this distribution

E Y E Geo

The number E Y tends fast to one as matrix density decreases

E Y

even though the we are considering sequences of unb ounded length For a comparably high

densityof cf Table we would exp ect comparisons for only

and for only comparisons

Prepro cessing

Iterated Intersection of Finite Random Sequences The innite sequence mo del

of the previous paragraph is not suited for an analysis of iterated intersections of se

quences b ecause under any reasonable assumptions a nite numb er of such sequences will

have an empty intersection with probability zero Thus we mo dify our mo del to deal with

nite sequences We lo ok at the sequences as the result of m indep endent rep etitions of a

exp eriment where the one has probability Formally this mo del can b e stated as

m fg

f g Bi

the mfold pro duct of the mo del f g Bi that describ es a single exp eriment here

Bi denotes the binomial distribution with parameters m and This nite mo del is of

course sub ject to the same criticism as its innite brother

To analyze the sequence intersection algorithm consider the following random experiment

Initialize an index set R as R f mg draw one sequence from f g after the other

at random and up date the set R byintersecting it with the sequences supp ort this pro cess

is continued forever Let a random variable X count the numb er of sequence intersections

until R b ecomes empty for the rst time

Lemma Iterated Intersection of Random Sequences

fg m

Bi b e a probabilityspace Let further X be a Let and f g

m m

random variable that returns for an innite numb er of randomly drawn sequences a a

fg k m

Bi the smallest number k such that supp a Then from f g

m i

E X m m

Pro of The rst step to compute E X is to note that the probabilityfor k sequences to

k k

have a common one in some place is not to have a common one in some place is

k m

to have empty intersection in all places is and the probabilityfor k sequences to

have nonemptyintersection in some of their m places is

k m

P X k k N

The exp ectation can now b e computed as

kP X k E X

m m

X X X

k m m k m

P X k

k k k

X X

k m k

m m m

k k

Here the inequality

m k m m k

follows from applying the mean value theorem to the function f R R t t

Considering the term as a constant we arrive indeed at ab out m sequences

that havetobeintersected Note that this number does not countthenumber of operations

in the iterated sequence intersection algorithm but the number of intersections

An Algorithm for Set Partitioning

Empty Columns Empty Rows and Row Singletons

Reductions P P and P are trivial and there is not muchtosay ab out their implementation

Tondempty columns it is enough to go once through the ob jective and through the cnt

array of the matrixs column representation which can be done in O n time Empty rows

are identied analogously in O mtime The neighb ors of a row singleton j are identied as

follows Scan column j each nonzero entry identies a row and all entries in this row except

for the singleton j itself denote neighb ors of the singleton and can b e eliminated Note that

a nonzero in a row is used at most once to identify the neighbor of arow singleton ie the

routine has linear amortized running time over all passes

Dominated Columns Duplicate and

Elimination of duplicate columns is a striking example for the eectiveness of even very

simple prepro cessing rules Table that gives statistics on the success of BCs individual

non LP based prepro cessing subroutines when applied to the Homan Padb erg

airline crew scheduling test set in Pass many passes shows the impact of this simple

reduction A quick glance at the table is enough to see that removing duplicate columns is

the most signicant prepro cessing op eration in terms of reduction in the numb er of nonzeros

and columns but not in rows of course One reason for this was already mentioned earlier

The acs problems as many other real world set partitioning instances were set up using

automatic column generation pro cedures that pro duce the same columns more than once A

second reason is that identical columns can very well corresp ond to dierent activities In

airline crew scheduling for example two rotations may service the same ight legs but on

dierent routes at dierent costs

The implementations of the literature seem to identify duplicate columns by comparisons of all

pairs of columns enhanced by hashing techniques Homan Padb erg compute a hash

value for each column quicksort the columns with resp ect to this criterion and compare all

pairs of columns with the same hash value The hash value itself is the sum of the indices of the

rst and the last nonnull entry in a column Atamturk Nemhauser Savelsb ergh use

the same algorithm but a more sophisticated hash function They assign a random number to

eachrow and the hash value of a column is the sum of the random numb ers corresp onding to

its nonnull entries Both pro cedures do in the worst case a quadratic numb er of comparisons

of two columns

BCs algorithm do es an exp ected number of O n log n comparisons by simply quicksorting

the columns lexicographical ly We remark that this strategy is particularly easy to implement

calling eg the Clibrarys qsortfunction In practice one can slightly improve the run

ning time by applying some linear time presorting op eration to the columns using eg some

hashing technique BC puts the columns into buckets according to the column cnt see

Subsection ie the number of nonnull entries and sorts the individual buckets as

describ ed ab ove

To estimate the expected running time of BCs quicksorting pro cedure we resort to Lemma

that states that in a certain uniform probabilistic mo del the exp ected num ber of op erations

to compare two random columns is constant for b ounded matrix density Since uniform dis

tribution of the sorted items in the partitions is an invariant of the quicksort algorithm this

results in an exp ected complexity of the complete pro cedure for removing duplicate columns

of O n log n op erations

Prepro cessing

Original P P

Name P Pass P P P P

Rows Cols Rows Cols Rows Cols

Table Analyzing Prepro cessing Rules

An Algorithm for Set Partitioning

Elimination of dominated columns is a generalization of removing duplicate columns The

latter reduction is the sp ecial case where the set J is restricted to contain just a single

member The main diculty in implementing the general rule is of course to nd this set J

in an ecient way The only known algorithm seems to be enumeration which is out of the

question even for medium sized problems This and the already mentioned prop erty of the LP

relaxation to keep dominated columns at zero values in optimal solutions anyway can explain

the apparentlack of implementations of this rule in the literature and the prepro cessor of our

algorithm BC do es also not search for dominated columns We remark that there are heuristic

pro cedures for the set covering variant of the reduction see Beasley

Duplicate and Dominated Rows

Removing duplicate rows of a matrix is equivalent to removing duplicate columns from

the transp ose Hence the implementation of this op eration is governed by exactly the same

considerations as for the columns The probabilistic analysis carries over as well since it

assumes only the indep endent random o ccurrence of ones in the matrix with probability

equal to the density The result is an exp ected number of O m log m index comparisons to

remove all duplicate rows

This favorable running time do es not entirely showupin our computations for the acs test

set The reason is that these instances come ordered in a staircase form with sequences

of consecutiv e ones in the rows which increases the probabilityof common nonzeros in two

rows This do es not t with the analysis and leads to an increase in running time of the

pro cedure

Figure Bringing Set Partitioning Problem nw into Staircase Form

A simple way out of this problem would be to p ermute rows and columns randomly but

staircase form also has its advantages elsewhere Since removing duplicate rows is not a

b ottleneck op eration we opted to leave the matrices as they are and employ more elab orate

presorting techniques instead We use a twolevel hashing rst assigning the rows to buckets

according to the numb er of nonzero entries as we did for the columns and then sub dividing

these buckets further into subbuckets of rows with the same sum of nonzero indices The

using shakersort an alternating bubblesort with almost individual subbuckets are sorted

linear exp ected running time for small arrays for small buckets elements in our

implementation and quicksort else With this tuning removing duplicate rows takes ab out

the same time as removing duplicate columns

As was already p ointed out earlier removing rows from a set partitioning problem is partic

ularly advantageous for branchandcut solvers b ecause it reduces the size of the LP basis

which has a quadratic impact on parts of the LP time like factorization For this reason

extending the removal of duplicate rows to dominated rows is of signicant interest Use of

the latter reduction is rep orted by Homan Padb erg and Atamturk Nemhauser

Savelsb ergh

Prepro cessing

BCs pro cedure to remove dominated rows is based on the sequence intersection algorithm of

Subsection It exploits the simple observation that the set of rows that dominate some

row A can b e expressed as the intersection of the columns in this row

supp A

j supp A

The metho d is simply to compute this set by intersecting the columns iteratively stopping

as so on as either r is the only row left in the set or when there is exactly one additional last

candidate row that is compared directly to r

We will argue in the next paragraph that one can exp ect the stopping criterion in this pro ce

dure to apply after ab out intersections of columns Considering ab out nonzero elements

in each column and if necessary one more see Subsection in the nal comparison of

two columns this results in an exp ected O op erations per row Doing this m times for

m ro ws we exp ect to remove all dominated rows in O m or if one likes this b etter in

O m steps

To estimate the numb er of column intersections until the stopping criterion applies we will

make use of Lemma We claim that E X is an upp er b ound on the exp ected num

berofintersections in the column intersection algorithm To see this note that all columns

considered in the algorithm have an entry in row r but their remainders are distributed

m fg

according to the mo del f g Bi for one row less namely row r

In this m row mo del E X counts the number of intersections until no row is left

which corresp onds in the original mrow mo del to the number of intersections until only

row r is left This ignores of course the p ossibilityto stop earlier if jR j in the mrow

hence E X is an upp er bound on the number of columns considered by mo del and

the algorithm

Row Cliques

Elimination of columns that extend a row clique seems to have b een used for computation by

Chu Beasley Using the same rules as Homan Padb erg PPP P

and P otherwise they rep ort a slightly bigger reduction in problem size A straightforward

way to implement the rule would b e to tentatively set eachvariable to one all its neighbors

to zero and check whether this contradicts some equation This algorithm requires however

one row scan for each nonzero element of the matrix This is not acceptable and as far as

we know nob o dy has suggested a b etter metho d to implement this reduction But we will

argue now and give some computational evidence that an exact implementation of this rule

is not worth the eort

Wewould exp ect from the analysis of the sequence intersection algorithm in Subsection

that the probability of a column to intersect many other columns in a row is extremely small

such that the chances to eliminate such a column are at b est questionable This argument

can b e made more precise using the probabilistic mo del of Subsection As we computed

in the pro of of Lemma ibidem the probabilityfor k columns to intersect in some of their

k m

m rows is P X k This probability which increases with larger values

and larger m and decreases with larger k resp ectiv ely is almost zero for the applications that

wehave in mind Considering unfavorable settings like a rather high densityof and

a comparably large number m of rows cf Table and a tiny row clique of just

columns this number is

An Algorithm for Set Partitioning

For this reason and for the sakeofspeedBC implements a heuristic version of P that consid

ers only rows with at most some constantnumber M of entries in our implementation

The rule is denoted byP and the eect of P on the Homan Padb erg test set

can b e seen from Table To satisfy our curiositywe have tested the full rule P as well

Applying P instead of P resulted in more rows and more columns removed by

the prepro cessor at the exp ense of several days of computation time

BCs implementation of rule P is based on the formula

j supp A

j supp A j supp A ssupp A

r r

Here the neighb ors of all columns in row r are determined byintersecting the neighbor sets

j of the individual columns and these are computed by scanning all corresp onding rows

The complete routine determines for some given matrix Aallrows with at most M nonzeros

and applies the ab ove pro cedure to each of these rows

To compute the expected complexity of this algorithm we consider again the probabilistic

mo del of Subsection that lo oks at eachrowofthem n matrix A as the result of

n indep endent exp eriments with a probabilityof for one If the random variable Y

counts the numb er of nonzeros in a row wewould exp ect P to take

O mP Y M M mn O Mn P Y M

n n

op erations The rst term mP Y M in this expression is the exp ected number of rows

with at most M nonzeros Each of these M nonzeros term two corresp onds to a column

with m entries on average and for each of these entries we have to scan arowwithabout

n entries

n M M

Arguments in the next paragraph suggest that P Y M M e n

This results in a total of

n M M

O M n e n

exp ected op erations to p erform P For a numerical example consider unfavorable pa

rameters of M m n the result is less than

The upp er b ound on the probability P Y M can b e computed as follows

i ni

P Y M

assuming n M

mnm

M m

nm M M

M e n

n M M

M e n

Prepro cessing

Parallel Columns

Elimination or merging of paral lel columns has b een used by Homan Padb erg The

rule requires some book keeping to be able to undo the merging of columns into comp ound

columns once a solution has b een found note that rep eated merging can result in comp ound

columns that corresp ond to sets of original variables BC do es not implement this rule and

we do not analyze it here

Symmetric Dierences

The symmetric dierence rule P is particularly attractive b ecause it leads to the elimination

of both rows and columns An implementation based on checking all triples of rows is a

disaster but the column intersection technique of Subsection canbeusedto design an

ecient pro cedure The algorithm in BC computes for each row s the column j with the

Nowwe distinguish two cases smallest supp ort

i Column j is supp osed to be contained in the symmetric dierence of row s and some

not yet known row t The only p ossible rows to cover the symmetric dierence are the

rows in supp A nfsg For eachsuchrow r thepotential rows t are limited to the set

supp A

isupp A nsupp A

s r

that agree with s on the columns that are not covered by r This set is computed using

iterative column intersection and each of the resulting candidates t is checked

ii Column j is supp osed to b e contained in the intersection of row s with some row t s

clearly s supp A For each such row t the symmetric dierence suppA A is

j s t

computed and a row r covering this dierence ie from the set

supp A

isuppA A

s t

is determined by means of iterative column intersection

A heuristic estimate of the running time of this pro cedure is as follows In case i we exp ect

to consider a column with less than the mean of m nonzeros For each nonzero we apply

the column intersection algorithm which takes O op erations and yields less than O

candidate rows t For each of these candidate rows we scan three rows which takes O n

op erations We would thus exp ect that p erforming i once for each of m rows takes a total

of O m n O m nm op erations In case ii we consider the same

column with exp ected nonzeros For each of these entries we scan tworows to compute a

symmetric dierence whichtakes O n op erations Then the column intersection algorithm

with O op erations is applied and eventually candidate rows checked taking another

O n op erations This results in O m n n overall op erations which is

of the same order as in the rst case Thus the total exp ected numb er of op erations in this

pro cedure is an acceptable

O m n O n

The same technique could also b e used to implement the generalization P of this rule but

unfortunatelythiswas not done for the prepro cessing mo dule of BC

An Algorithm for Set Partitioning

Column Singletons

Elimination of rows and columns using column singletons is a sp ecial case of a more general

substitution op eration This technique works as follows Consider an integer program

T n

min w w x Ax b Cx d l x u x Z

with an ob jective that contains a constant oset term w some equations inequalities and

lower and upp er b ounds p ossibly on the variables Without loss of generality we can

assume a to b e p ositive and bring the rst equation into the form

a a x x b a

i i

x by Gaussian elimination in the ob jective the other This equation can b e used to eliminate

constraints and the b ounds The result of this op eration is one variable x and one con

straintA x b less p otential l l in the equations and inequalities and a transformation

of the two b ounds l x u and the integrality stipulation x Z into the form

X X

b a u a x b a l and b a a a x Z

i i i i

i i

Sometimes these constraints will be redundant One restrictive but relevant and easily de

tectable case is when the transformed integrality stipulation on the right of holds b ecause

mn m

the equations Ax b have integer data ie A Z and b Z and a ie the

pivot is one and there is no division in the Gaussian elimination and when in addition the

transformed b ounds are redundant b ecause

X X

b u min fa l a u g max fa l a u gb l

i i i i i i i i

i i

Under these circumstances the substitution results in a reduction in the number of rows and

columns of the program and we sp eak of preprocessing by substitution This technique is

widely used in LP and MIP solvers eg in CPLEX To control ll implementations

generally restrict substitution to columns with few entries like singleton columns or to rows

with few entries like doubleton rows see Bixby

An obstacle to the application of this rule to set partitioning problems is that the b ound

redundancy criterion is computationally useless in this application Namely assum

ing that all xed variables have already been removed earlier condition reads

j suppA e j This can and will hold exactly for the trivial case of a doubleton row

Another criterion is thus needed for set partitioning problems and

A A e

s r

for another row s r as suggested in rule P is a suitable choice to guarantee

Row s can b e identied by column intersection and this yields a running time of at most

O m nO n

op erations Atmostm singletons can b e eliminated each candidate requires one application

of the column intersection algorithm with O op erations and the resulting candidate row

is checked and substituted into the ob jective in O n op erations

Prepro cessing

Note that the result of a sequence of substitutions is indep endent from the elimination order

but the amountofwork is not The three equation example in Figure illustrates how this

is meant In the example column singleton x can be eliminated from equation After

the rst row and column have been deleted x can be eliminated using equation and

nally x from Doing the substitutions in this order x from x from and

x from pro duces no ll Given the original matrix A and this ordered substitution

history list one can repro duce a Ppro cessed problem in time prop ortional to the number

of nonzeros in the substitution equations by substituting these equations in the given order

into the ob jective and by eliminating rows and columns Using some other order leads to

additional work For example substituting in Figure for x rst using pro duces two

nonzeros in equation at x and x and continuing to substitute in any order results in a

worst p ossible ll

x x

x x x

x x

Figure Eliminating Column Singletons in the Right Order to Avoid Fill

This phenomenon can b ecome relevant in a branchandcut context on two o ccasions First

when a solution to a prepro cessed problem has b een found and substitutions have to be re

versed this should be done in reverse order of the substitution history by computing the

values of the substituted variables from the corresp onding original equations note that this

do es in general not work with some other elimination order And second when a prepro cessed

subproblem in the searchtree has to be reproduced BC do es not store the ob jectives of pre

pro cessed subproblems b ecause this would require an arrayofO n double variables at each

no de Instead the ob jective is recomputed from scratcheach time Doing this without mak

ing any substitutions in the matrix requires a zero ll substitution order The consequence is

to store the order of column singleton substitutions on a history stack

Analword has to b e said ab out the impact of substitution on the ob jective function and the

solution of the LP relaxation Many set partitioning problems from scheduling applications

have nonnegative ob jectives and so do the acs problems Substitution destroys this prop erty

by pro ducing negative co ecients Unexp ectedly the LP relaxations of problems that were

prepro cessed in this way b ecome dicult to solve probably b ecause the start basis heuristics

do not work satisfactory any more But fortunately there is a simple way out of this dilemma

The idea to counter the increase in LP time is to make the ob jective positive again by adding

suitable multiples of the rows BCs pro cedure implements the formula

w w x Ax

where

jw jj supp A j r m max

j j r j supp A w

The impact of rule P on the acs test can be read from Table A nice success is that

problem nw can be solved by prepro cessing with this rule P has also proved valuable in

dealing with set packing constraints see the vehicle availability constraints in Chapter in

a set partitioning solver Transforming such inequalities into equations intro ducing a slack

variable pro duces column singletons that can p otentially b e eliminated

An Algorithm for Set Partitioning

Reduced Cost Fixing

Reducedcost xing is another example of a strikingly simple and eective prepro cessing op era

tion Wedraw the readers attention to Table where a comparison of the Presolved and

Presolved LPbased columns indicates that reduced cost xing based on the knowledge

of a go o d upp er b ound from the primal heuristic accounts for a reduction in the number of

columns and nonzeros of one order of magnitude

Probing

Probing a rule that wehave not completely sp ecied b elongs to a group of expensive prepro

cessing op erations in the sense that they require the exact or approximate solution of linear

programs There is additional information gained in this way that makes these op erations

powerful P is for example stronger than P but there is of course a delicate tradeo

between time sp ent in prepro cessing and solving the actual problem

An implementation of probing by ten tatively setting variables to their b ounds can be done

with p ostoptimization techniques using advanced basis information Having an optimal basis

at hand one sets one variable at a time to one of its b ounds and reoptimizes with the dual

simplex metho d after that one reloads the original basis and continues in this way This

metho d has the disadvantage that there is no control on the amount of time sp ent in the

individual LPs Some control on the computational eort is gained by limiting the number

of simplex iterations in the p ostoptimization pro cess at the cost of replacing the optimal LP

value with some lower b ound If the iteration limit allows only few iterations this oers the

additional p ossibilitytoavoid basis factorizations using an eta le tec hnique In each prob e

the basis is up dated adding columns to the eta le when the iteration limit is exceeded or the

problem solved the original basis is restored by simply deleting the eta le This technique

is implemented in CPLEX but despite all these eorts probing is still exp ensive

BC uses probing of variables in its default strong branching strategy cf Bixby p ersonal

communication Some set of candidate variables for probing are determined the most

fractional ones each of these is prob ed dual simplex iterations deep and any p ossible

xings are carried out the remaining b ound information is used to guide the branching

decision

Pivoting

We have seen in the intro duction to this section that prepro cessing is an eective to ol to

reduce the size of a given set partitioning problem and that techniques of this sort can help

to solve these IPs faster There is no reason to b elieve that this do es not also work in the

same way for the subproblems created by a branchandb ound algorithm Rather to the

contrary one would exp ect iterated preprocessing on subproblems to be even more eective

since subproblems contain additional xings due to branching decisions and the lower b ound is

b etter To exploit this information one would like to prepro cess not only the original problem

formulation but also subproblems repeatedly throughout the branchandb ound tree

of the basis Instead LPbased metho ds on the other hand liveonmaintaining dual feasibility

of solving an LP from scratch each time a variable has been xed in a branching decision

or a cutting plane has been added the dual simplex metho d is called to reoptimize the LP

starting from the advanced basis obtained in the preceding optimization step

Prepro cessing

These two principles rep eated problem reduction and maintenance of a dual feasible basis

can get into conict Reductions that do not interfere with a dual feasible basis are

i Eliminating nonbasic columns

ii Eliminating basic rows ie rows where the asso ciated slack or articial variable is basic

i it do es obviously neither aect the basis itself nor its dual feasibility ii is p ossible since

the multiplier dual variable asso ciated to a basic row r is zero But then the reduced costs

T T T T

w w A w A are not aected byremoving row r and moreover if A

s s B

s r

denotes the matrix that arises from the basis matrix A by deleting row r and column e

B r

this reduced basis A is dual feasible for the reduced problem

Rule ii can b e slightly extended with a pivoting technique to

iii Eliminating rows with zero multipliers

The metho d is to reduce iii to ii by p erforming a primal pivot on a where row r with

is supp osed to b e eliminated and j the unit column corresp onding to its slackarticial

primal pivot means that the slackarticial column e is entering the basis As w

r r j

this pivot will b e dual degenerate We are interested here in the case where row r is known to

b e a linearly redundant equation then its articial variable is zero in any feasible solution

and the pivot will also be primal degenerate This inpivoting pro cedure was develop ed

by Applegate Bixby Chvatal Co ok for the solution of large scale TSPs and is

implemented in CPLEX V and higher versions

One p ossible strategy for iterated prepro cessing in a branchandcut algorithm is thus the

following Apply the prepro cessor as often as you like and eliminate rows and columns using

iiii doing inpivoting prior to the actual elimination of rows where necessary If a basic

column was eliminated or xed to one by the prepro cessor change its b ounds but leaveitin

the formulation and do also not removerows with nonzero multipliers form the formulation

even if the prepro cessor detected their redundancy If to o much garbage accumulates

eliminate everything discard the useless basis and optimize from scratch

One might wonder whether it is at all p ossible that redundant rows can have nonzero mul

tipliers Do not all row elimination rules except for the column singleton rule P after

elimination of certain columns result in sets of duplicate rows where at most one represen

tative can have a nonzero multiplier The following simple example shows that this is not so

and why Consider the set partitioning problem

min x x

c x x y

c x y

x x f g

Here the variables y and y denote the articial variables of the constraints c and c

resp ectively The rst two columns of the constraint matrix corresp ond to the variables x

and x and constitute an optimal basis for the corresp onding simplex tableau reads

min y y

c x y y

c x y

x f g x

The values of the dual variables are b oth nonzero

An Algorithm for Set Partitioning

Supp ose that in this situation a prepro cessor investigates formulation and nds out that

variable x can be eliminated Consider the example as a part of a bigger problem and

ignore the p ossibilityto solve the problem by xing also x to one Eliminating x in the

prepro cessing data structures not in the LP results in two identical rows candc Supp ose

the prepro cessor nds this out as well and suggests to eliminate one of them But whether

we try to eliminate c or c neither of these suggestions is compatible with dual feasibility

of the basis and we can not eliminate rows and columns that weknow are redundant Since

linear dep endent and much less duplicate rows can not be contained in a basis there must

b e some xed variable in the basis Clearly there must b e alternative optimal bases that do

not contain one or some or all xed variables We suer from primal degeneracy

The degeneracy phenomenon that wehave just describ ed do es not only app ear in theorybut

is a ma jor obstacle to the solution of set partitioning problems by branchandcut Unexp ect

edly it turns out that for the airline test set often almost half of the basis matrices consist

of xed variables blo cking the same number of rows from p ossible elimination It is clear

that a larger number of rows and a larger basis has a negative impact on LP time

This problem can b e overcome bya novel outpivoting technique that forces xed variables to

leave the basis The metho d is to p erform one dual pivot with the xed basic variable leaving

the basis allowing slacksarticials to enter As the leaving variable is xed this pivot is

primal degenerate but the dual solution changes and the entering variable is determined in

sucha way that optimality is reestablished ie by a ratio test

Outpivoting is available in CPLEX release V and higher version Its use to eliminate xed

variables from the basis allows for signicant additional problem reductions while at the same

time maintaining dual feasibility We remark that although the metho d is b est p ossible in the

sense that it requires just a single dual pivot for each xed basic variable outpivoting is not

cheap Table shows that of the total running time that our branchandcut algorithm

BC needs to solve the airline test set is sp ent in outpivoting column Pvt under Timings

And the numb er of outpivots exceeds the numb er of other pivots by a factor of ab out ve

The Prepro cessor

Combining the routines of the previous subsections yields the preprocessor of our set parti

tioning solv er BC The mo dule consists of kilobytes of source co de in lines

The mo dule do es not work on the LP itself but on a p ossibly smaller auxiliary representation

of the problem where reductions can be carried out no matter what the LP basis status is

The prepro cessor is called for the rst time prior to the solution of the rst LP All later

invocations involve pivoting to maintain the dual feasibilityof the basis First the basis is

purged bypivoting out xed variables from previous invo cations Prepro cessing starts with

reduced cost xing according to rule P Then the main prepro cessing lo op is entered that

of column oriented reductions calls in each pass all the individual rules First a couple

are carried out P row singletons and P row clique heuristic Then the matrix is

transp osed and roworiented op erations follow P duplicate rows P dominated rows

P symmetric dierence and P column singletons The matrix is transp osed again for

the next pass This lo op continues as long as some reduction was achieved When no further

reductions can b e achieved as many of the found ones as p ossible are transferred to the LP

Articials of redundantrows are pivoted in and redundantnonbasic columns and redundant

basic rows are eliminated from the LP The reader can infer from Table that the running

time for this mo dule is not a computational b ottleneck for the entire branchandcut co de

Separation

Branchandcut this term lists the two sources of p ower of the algorithms of this class

The second of these the computation of cutting planes aims at improving the quality of the

current LP relaxation in the sense that the lower b ound rises If this can be achieved it

helps in fathoming no des and xing variables by prepro cessing techniques provides criteria

for intelligent searchtree exploration and ideally pushes the fractional solution toward

an integral one This in turn can be exploited for the development of heuristics by trac

ing histories of fractional variables etc and there are certainly more of such practitioners

arguments in favor of cutting planes that are all based on the many algorithmically useful

degrees of freedom in as the name says a generic branchandcut metho d The theoretical

justication for the use of cutting planes is p erhaps even more convincing By the general

algorithmic results of Grotschel Lovasz Schrijver we know that p olynomial time

separation allows for p olynomial time optimization and even if we give here the dual simplex

algorithms reoptimization capabilities not to sp eak of the availability of suitable implemen

tations preference over the ellipsoid metho ds theoretical p ower there is no reason to b elieve

that not some of this favorable b eha viour will show up in co des of the real world And in fact

the number of implementations of this principle with successful computational exp erience is

legion see eg Caprara Fischetti for a surv ey

The separation routines for set partitioning problems are based on the relation

P AP A Q A

I I

between the set partitioning the set packing and the set covering p olytop e To solve set

partitioning problems we can resort to cutting planes for the asso ciated packing and covering

p olytop es Wehave already p ointed out in Section why the p olyhedral study of the latter

two b o dies is easier than the study of the rst and wehave also listed in the Sections

and manyknown typ es of valid and often even facet dening inequalities that qualify

as candidates for cutting planes in a branchandcut co de for set partitioning problems

But not only these classes are available General cutting planes suggest themselves as well

Gomory cuts liftandpro ject cuts see Balas Ceria Cornuejols or Martin

Weismantel s feasible set cuts

We have selected only a small number of them for our implementation Clique inequalities

b ecause they give facets are easy to implement numerically stable only co ecients and

sparse cycle inequalities for the same reasons and b ecause they can b e separated exactly and

the aggregated cycle inequalities from the set packing relaxation of the set covering problem

of Section b ecause wewanted to evaluate the computational usefulness of our aggregation

technique These cuts are all simple but as the duality gaps in real world set partitioning

problems are usually quite small there is some justication for a strategy that opts for

whatever one can get in a short time

We discuss in the following subsections the individual routines of our separation mo dule

All of the pro cedures work with intersection graphs that we intro duce in Subsection

Separation and lifting routines for inequalities from cliques are treated in Subsection

for cycles in Subsection and for aggregated cycles in Subsection Aword on our

strategies to call these routines can b e found in the following Section

An Algorithm for Set Partitioning

The Fractional Intersection Graph

All of our separation routines will be combinatorial algorithms that work on intersection

graphs Namelywe lo ok for our set packing inequalities on subgraphs of GA the intersection

graph of the set packing relaxation and weidentify aggregated cycle inequalities on subgraphs

of the conict graph GA that is asso ciated to the aggregation

Aquick calculation is enough to see that it is completely out of the question to set up GA

completely and much less GA and even if we could do this it is very unlikely that we could

makeany use of these gigabytes of information But luckily it follows from the nonnegativity

of all nontrivial facets of set packing p olytop es and the connectedness of their supp ort

graphs is well known and was mentioned for example in Homan P adb erg that

one can restrict attention to the fractional parts

GAF GA and GAFGA

F F

of these structures for separation purp oses These graphs are the fractional intersection

graph and the aggregated fractional intersection graph resp ectively As there can b e at most

as many fractional variables as is the size of the basis as is the number of equations of the

LP relaxation this reduces for typical real world set partitioning problems like the airline

instances the numb er of no des from ten to hundred thousands in GAtosomehundreds in

bytwo to three orders of magnitude and the numb er of edges even more This is not GA

so for the graph GA which is exp onential by construction We cop e with this diculty

in a heuristic way by using only some subgraph of GA Note that the ab ove mentioned

connectedness of the supp ort graphs of facets makes it p ossible to restrict separation to

individual connected comp onents of GA and of GA

F F

Separating on the fractional variables only has the disadvantage that the resulting cutting

planes have a very small supp ort in comparison to the complete set of variables One way

to counter the stalling eects of p olishing on a low dimensional face of the set partitioning

p olytop e is to extend the supp ort of cutting planes by lifting Our overall separation strategy

will b e to reduce the eort to identify a violated inequalityasmuch as p ossible byworking on

fractional intersection graphs and we enhance the quality of whatever wewere able to obtain

in this rst step a posteriori by a subsequent lifting step

We turn now to the algorithmic construction of the fractional intersection graph We treat

only GA and do not discuss here howwe set up a subgraph of GA b ecause this is so

F F

intimately related to the separation of aggregated cycle inequalities that it is b etter discussed

in this context in Subsection

that we have implemented in BC sets up a new column intersection graph The procedure

GA after the solution of every single LP ie GA is constructed on the y as

F F

Homan Padb erg say Our routine uses two copies of the matrix A one stored

in column and the other in row ma jor format A can b e extracted from the column ma jor

representation of the global matrix A in time that is linear in the number of nonzeros of

A Next we compute the neighbors of each column j F by scanning its rows and store

the result in a forward star adjacency list see eg Ahuja Magnanti Orlin Under

the assumptions of Subsection we exp ect that this will takeaboutO jF j op erations

on average fast enough to just forget ab out We do not use a pro cedure to decomp ose

GA into two connected comp onents

Separation

Clique Inequalities

Wehave already mentioned in the intro duction of this section whywe use clique inequalities

as cutting planes in our branchandcut co de BC This class yields facets it is easy to come up

with separation and lifting heuristics and such inequalities are sparse and p ose no numerical

diculties One must admit however that these app ealing prop erties are strictly sp eaking

outmatched by the unsatisfactory theoretical b ehaviour of these simple cutting planes Clique

separation is not only NP hard see Garey Johnson but even worse this class is

contained in p olynomial separable sup erclasses like orthogonality inequalities or matrix cuts

One could argue somewhat around the rst dicultynamelywehave implemented an exact

clique separation routine as well and found that even without any tuning our heuristics

already found nearly every violated clique inequality there was and it is a little thing to tune

the heuristic routines such that containment b ecomes equalit y But we feel nevertheless that

the ab ove arguments show that it is not the rightway to comp ensate the conceptual weakness

in clique inequality separation by additional computational eort

Our branchandcut co de BC go es thus to the other extreme and concentrates on the compu

tational advantages of heuristic clique detection by using only simple separation and lifting

routines We compute violated inequalities with a row lifting and a greedy heuristic and a

semiheuristic the meaning of this term will b ecome clear in the description of this metho d

recursive smal lest last RSL pro cedure and we lift the cutting planes that they return with

tailor made pro cedures that t with the separation routines philosophy these statements

have b een evaluated in computational exp eriments These separation and lifting routines

are describ ed in the next paragraphs

Row Lifting Homan Padb erg The idea of this separation routine is to

exploit the knowledge of those cliques that are already enco ded in the rows of the matrix A

to design a very fast pro cedure The details are as follows

One considers each row A of the matrix A that consists of the columns of A with

rF F

fractional variables in the current solution x in turn note that the sum over the fractionals

in a row is either zero there are no fractional variables b ecause some variable has a value of

one or one

A x f g r m

In the latter case this row induces a minimal clique Q supp A such that the clique

t LP solution x If one additional fractional inequality x is tight for the curren

j Q

variable can be lifted sequentially into Q a violated clique inequality is detected Lifting

more fractional variables increases violation and one can lift some additional variables with

zero values in the end as well to extend the cuts supp ort Hence the pro cedure has three

steps i Determining the core clique Q supp A ii sequential lifting of fractional

variables into the core and iii supplementary sequential liftings of zero variables

Here are some implementation issues While i is clear one can come up with numerous

strategies for the lifting steps ii and iii The metho d that we have implemented in BC

opts for speed b ecause we do not exp ect to nd many additional neighbors of a part of

a matrix row clique that is usually of substantial size a philosophy that ts with the

idea b ehind the row lifting metho d and that is supp orted by the probabilistic results of the

previous Section and by our computational exp eriments For each of the steps ii and

iii we set up a list of candidate variables that we arrange in a xed lifting order and this

An Algorithm for Set Partitioning

candidate sequence is used for every row In step ii the candidate set consists of some

constantnumber k of the fractional variables with the largest x values weusek in

F F

our implementation which are tried greedily in order of decreasing x values and another

constantnumber k of zero variables we use k for step iii that we simply select at

L L

random

Turning to the exp ected running timewe note that one sequential lifting of a variable x can

b e done bychecking whether all variables in the current clique Q are neighbors of j Using a

forward star representation of GA thistakes O j j j O jF j steps where jF j denotes

the numb er of fractional variables in the current LP solution x Doing this k k O jF j

F L

times for m rows results in a total of O mjF j op erations for this routine whichisasfast

as one could p ossibly hop e

The apparent disadvantage of the metho d is however that the cutting planes that one com

putes with sucha technique do by construction resemble much subsets of rows with a small

extension here and there Generally sp eaking the row lifting clique separation routine is

a go o d starting metho d in the initial phase of a branchandcut run and yields reasonable

results there it is less useful in later stages of the computation

The greedy metho d is certainly the most obvious and simple to im Greedy Algorithm

plement separation strategy that one can come up with and our branchandcut algorithm BC

also uses a clique detection metho d of this typ e

Our routine is implemented in the following way The greedy criterion is to go for a most

violated clique inequality and it makes sense to do so by considering the fractional variables

in order of decreasing x values where x denotes the current fractional solution

x where f g F x x

jF j

Our greedy do es now jF j trials one for each fractional seed variable x In trial j we

initialize a clique Q f g that will hop efully be grown into the supp ort of a violated

clique inequalityandtrytoliftinto Q all variables x x of smaller x value in this

jF j

order The motivation b ehind this is to give variables with small x values also a chance

to foster a violated clique We do not restrict the numb er of fractional lifting candidates this

time b ecause we exp ect for familiar reasons that the cliques that we can compute in this

Note that this is dierent from row lifting where we start a way will not be very large

priori with a large clique This inspires the dierent lifting philosophy that we should at

least lift such small cliques reasonably to put it nonchalantly But how can we get a large

extension when all our probabilistic analyses and computational exp erience indicates that we

can not obtain it sequentially Our idea is to use the large cliques that we already know and

to do a simultaneous lifting with matrix rows similar to the row lifting separation routine

Namely we do the following Given some fractional clique Q we determine its common

neighbors Q j notethatitisnotclever to compute this for a large clique but

j Q

no problem for a small one and then we lo ok for the largest intersection Q supp A of

this set with the supp ort of some row r this set is added to Q

Lo oking at running times we have again that one sequential lifting of a fractional variable

takes O jF j op erations Lifting at most jF j variables in the greedy clique growing phase

results in O jF j steps The common neighb ors of at most jF j memb ers of such a clique can b e

determined in O jF jm n steps using the matrix As row representation not the complete

intersection graph GAwhichwe did not set up and the maximum intersection of this set

with a matrix rowinO m n steps which is smaller Assuming O jF jm n O jF j

Separation

and doing all of these steps jF j times once for each of the seed variables amounts to a

total of O jF j mn exp ected steps whichdoes not lo ok very go o d But our analysis is

a very conservative estimate b ecause the exp ensive simultaneous lifting step is only called

when a violated inequality is found which unfortunately is not the case for every starting

candidate The metho d can b e tuned further using obvious break criteria based on the tail

sums

j jF j x x

jF j

that make the routine bail out whenever there is no more chance of nding a violated clique

With this and other improvements of this typ e one obtains a separation pro cedure inequality

that displays a reasonable b ehaviour in computational practice

Recursive Smallest First One of the most p opular branchandb ound approaches to the

maximum weight clique problem is based on the recursion

max x Q max fx max x Q max x Qg

Q clique in G Q clique in Gj

Q clique in G j

Here G is some graph with no de weights x and j one of its no des The rst successful

implementation of is as far as we know due to Carraghan Pardalos and since

then this branching rule has turned into the progenitor of a large family of algorithms that

dier by no de selection and clever b ounding criteria that try to reuse information that is

computed once as often as p ossible

Recursive smal lest rst RSF is one member of this class It uses the sp ecial branching

strategy to select in each step a no de j that attains the minimum degree in the current graph

The idea is obviously that one of the two subproblems namely the one on the neighb ors of j

ie on the graph G j will hop efully b e small and can b e fathomed or solved fast For

fathoming we can develop simple criteria in terms of sums of no de weights of the current

graph And the subproblem can surely b e solved fast if the number of no des in the current

graph is small say smaller than some constant k When such circumstances sup ervene in

every sub division step the RSF algorithm solves the maximum weight clique problem to

proven optimality in time that is p olynomial of order k The worst case running time is

exp onential however

The observations of the previous paragraph suggest a simple waytocombine under favorable

conditions the advantage of RSF a certicate of optimality with a p olynomially b ounded

running time The idea is to turn the algorithm dynamical ly into a heuristic whenever we are

ab out to walk into the complexity trap Namely we pursue the following strategy We use

in principle the generic RSF algorithm as describ ed ab ove but whenever the current graph

has more than k no des and our fathoming criteria fail we solve the asso ciated subproblem

heuristically We call sucha hybrid metho d with b oth exact branchandb ound and heuristic

comp onents a semiheuristic A scheme of this typ e has the advantages that it i is able

to exploit some structural prop erties of the graph namely to reduce it systematically by

cutting o low degree parts ii it allows to control the tradeo b etween exactness and sp eed

by tuning the parameter k and iii it sometimes even proves optimality of the result

k When the current In our implementation of the RSF metho d we set the parameter

graph has less than this number of no des we determine the maximum clique by complete

enumeration The heuristic that we apply in subproblems that involve graphs with more than

k no des is the greedy pro cedure that wehave describ ed in the previous paragraph To nd a

An Algorithm for Set Partitioning

no de with smallest degree in each branching step we store the no des of the graph in a binary

heap that is sorted with resp ect to no de degrees For familiar reasons we do not exp ect

the RSF algorithm to return a large clique In this vein RSF has the avour of an improved

greedy algorithm Therefore we apply the same strategies to lift variables that haveavalue of

zero in the current LP solution Homan Padb erg describ e a similar implemen tation

of the RSF metho d

The running time of RSF is O jF j The time to compute lower b ounds is O jF j the

greedy heuristic takes O jF j enumeration takes O jF j and the heap up dates require

O jF j log jF j op erations nally lifting results in O jF jmn steps This gives a total

k k

running time of O jF j jF jmn whichwe assume to b e of order O jF j

Toevaluate the quality of the RSF metho d wehave implemented an exact branchandb ound

algorithm for the maximum clique problem as well It turned out that even without any

tuning RSF almost always pro duced a largest clique Our computational exp eriments showed

that the choice k was the optimal tradeo b etween sp eed and quality In fact with k

RSF pro duces alw ays the largest clique on the airline test problems For this reason and

b ecause of the arguments mentioned in the intro duction of this section we do not use the exact

branchandb ound algorithm for clique separation although this metho d is implemented

Cycle Inequalities

Cycle inequalities are the second separation ingredient in our branchandcut algorithm Like

the clique inequalities cuts of this typ e have small supp ort and they tend to have a nice

numerical b ehaviour only co ecients in unlifted versions An additional b onus is that

they can be separated in polynomial time with the GLS algorithm of Grotschel Lovasz

Schrijver We use this cycle detection algorithm in our branchandcut algorithm

The GLS algorithm works on a bipartite auxiliary graph B B GA that is constructed

from the fractional intersection graph GA V E as follows The no des of B are two

copies V and V of V There is an edge u v in B if and only if uv is an edge of G To each

suchedge u v we asso ciate the weight w x x where x is the current fractional

u v

LP solution Note that w

The main steps of the pro cedure are as follows One computes for each no de u V the

in B to its p endant u Each such path P interpreted as a set of no des shortest path P

u u

corresp onds to an o dd cycle C in G through u p ossibly with no de rep etitions The weight

of C is

w C w P x x jP jx P jC jx C

u u u u u u

u v

u v P

Hence

w C jC j x C jC j x C

u u u u u

Thus a path P in B with weight less than one corresp onds to a violated o dd cycle inequality

Conversely a shortest o dd cycle through a no de u corresp onds to the path P This proves

that the GLS algorithm solves the separation problem for cycle inequalities in p olynomial

time

Our implementation of the GLS algorithm computes the shortest paths P using Dijkstras

algorithm When the distance lab els of the no des are kept in a binary heap this results in a

jF j jF jjE jO jF j log jF j jF j here jE j is the number of running time of O jF j log

edges in the fractional intersection graph

Separation

Weuseanumb er of implementation tricks to make this metho d work in practice First note

that it is not necessary to set up the auxiliary graph explicitly b ecause adjacencies in B can

b e read o from the neighb or lists of GA The only place where the auxiliary graph shows

up explicitly is the heap where we have to store a distance lab el for each of the two copies

of anode Second one can exploit the sp ecial form of the distance function x x for

computing expressions of the form

distv min fdistv distu x x g

u v

required to that come up in the relab eling step The three arithmetic op erations that are

compute the term distu x x for every neighbor v of u can be reduced to one

u v

by a precomputation of the term distu x A minor sp eed up can be achieved by

turning double x values into integers this saves ab out of the running time Third

Dijkstras algorithm is a dynamic program As we are interested in paths of length smaller

than one only wecan fathom a no de as so on as its distance lab el distv attains avalue of

one or more Fourth note that the generic GLS algorithm computes the shortest path P for

every no de u GA Once this path P is computed for a particular no de u this no de

F u

can b e deleted from the graph without lo osing the exactness of the metho d This is correct

b ecause a most violated cycle inequality passes through the no de u or not In the rst case

the path P yields such a most violated inequality In the second case u is not relevant and

can therefore be removed Note that this elimination strategy has the additional advantage

that it tends to pro duce violated cycle inequalities with disjoint supp ort It also pav es the

way for a fth implementation trick that is based on a sp ecial ordering of the starting no des

for which we call Dijkstras algorithm We order the no des with resp ect to decreasing x

value

x x x

jF j

If we denote by G the graph GA f g obtained from GA by deleting the

i F i F

jF j

no des f g the starting no des for the previous i calls of Dijkstras algorithm

all edge distances in G satisfy

w x x x

u v

Any o dd cycle C in G must contain at least three such edges and wehave for its weight

w C jC j x jC jjC jx

v C

Thelastvalue in this sequence exceeds if and only if

jC j jC j x

ie we can stop computation as so on as the maximum This will be the case if x

x value drops b elow We compute with the GLS algorithm and these tricks paths P

that corresp ond to o dd closed walks in GA and extract from these a cycle without no de

rep etitions

Lifting o dd cycle inequalities is a bit more complicated than lifting clique inequalities

Let us rst turn to sequential lifting Note that it is not dicult to lift a constantnumber of

variables into a cycle We tried an implementation that do es this for the rst two variables

An Algorithm for Set Partitioning

such that in each of the two steps a maximum additional violation of x times the lifting

co ecient is achieved More fractional variables were lifted heuristically This sequential

metho d turned out to be slow taking more time than the separation of the pure cycles

Moreover it did not pro duce many nonzero lifting co ecients

Therefore a simultaneous lifting metho d was implemented This metho d identies for each

breaking edge ij in the cycle C arow r r in the matrix A such that fi j gsupp A

ij F r

ties arbitrarily These rows are used to compute a ChvatalGomory cut that can b e seen as

a lifting of the cycle inequalitythat corresp onds to C We add up the rows A divide by

two and round the co ecients down Exploiting sparsity this metho d can b e implemented

in O jC jn time and exhibits a satisfactory computational b ehaviour

One nal issue on cycle separation is that it is p ossible that a violated inequality can result

from a lifting of a pure cycle inequality whichisnottight We exploit this heuristically in our

routine by increasing the target length of the paths in the GLS algorithm form one to some

larger value in a dynamic and adaptive fashion dep ending on the numb er of cycle inequalities

found in the previous call

Aggregated Cycle Inequalities

The third class of inequalities that we try to separate are the aggregated cycle inequalities

of Section Recall that these inequalities stem from a set packing relaxation of the set

covering problem

Set packing inequalities tend to have the disadvantage of smearing the values of the LP

solution over their supp ort This tends to increase the number of fractional variables with

small values which has all kinds of negative impacts on the solution pro cess To counter

these eects one would like to use cutting planes for the set covering p olytop e that gather

some x value on their supp ort and prevent the LP solution from dilution Unfortunately

little algorithmically useful knowledge ab out such cutting planes is available This was our

motivation for the development of the aggregated cycle inequalities

Aggregated cycle inequalities are separated with the implementation of the GLS algorithm

that wehave describ ed in the previous subsection The only dierence is that the input graph

is a small subgraph G V E of the aggregated fractional intersection graph GA

which is of exp onential size The selection is guided by the desire to nd a subgraph of reason

able p olynomial size and with many edges uv with smal l weights x x w

u v uv

Such edges make it likely that cycles in G give rise to violated aggregated cycle inequalities

We do not lift aggregated cycle inequalities

I Our heuristic to generate the subgraph G is the following We generate two no des I and

for each row A of the matrix A Namelywe sub divide the supp ort of eachrow A into

iF F iF

two equal sized halves

I supp A I

with resp ect to a given fractional LP solution x ie we split in some way such that

A A x x and take V as the set of these halves

i I

I j i mg V fI

Two such no des u and v are in conict if their union contains some row of the matrix A

These conicts dene the edges of the graph G

Computational Results

We rep ort in this section on computational experiences with our branchandcut co de BCWe

intend to investigate the following questions

i Performance What is the p erformance of BC on a standard test set of set partitioning

problems from the literature The acs test set of Homan Padb erg

ii Branching versus Cutting Do cutting planes make a signicant contribution to the

solution of the problems in our test set

iii Aggregated Cycle Inequalities What is the eect of the aggregated cycle inequalities

Wehavechosen the airline crew scheduling problems of Homan Padb erg as our test

set see this reference for a thorough discussion of these instances b ecause they are publicly

available and well known to the community This makes it p ossible to compare our results

with those of the literature see eg Homan Padb erg Atamturk Nemhauser

Savelsb ergh and Chu Beasley

According to the guidelines of Crowder Dembo Mulvey and Jackson Boggs Nash

Powell for rep orting ab out computational exp eriments we state that all test runs

were made on a Sun Ultra Sparc Mo del E workstation with MB of main memory

running SunOS that our branchandcut co de BC was written in ANSI C compiled with

the Sun cc compiler and switches fast xO and that we have used the CPLEX

Callable Library V as our LP solver

The results of the following computational exp eriments are do cumented in tables that have

the following format Column gives the name of the problem columns its size in terms of

numb ers of rows columns and nonzeros These sizes are reduced by an initial prepro cessing

to the numb ers that app ear in the next three columns Columns and rep ort solution values

z is the value of the b est solution that the algorithm has computed The s in the succeeding

Gap column indicate that all of the problems have b een solved to proven optimality The

following columns give details ab out the branchandcut computation We list from left

to right the numb er of in and outpivots Pvt that are p erformed by the prepro cessor the

number of cutting planes Cut added the number of simplex iterations to solve the LPs

Itn the number of LPs solved LP and the number of branchandb ound no des BB

Running times as a p ercentage of the total time for these routines are contained in columns

Problem reduction PP pivoting Pvt separation Cut LPsolution LP and

primal heuristic Heu The last column gives the total running time in CPU seconds

If not explicitly stated otherwise all of our computations use the following default p arameter

settings and strategies for our co de BC We use a best rst search on the branchandb ound

tree the branching rule is strong branching cf Bixby p ersonal communication ie we select

a set of fractional candidate variables close to we try candidates x them tentatively

to and and p erform a couple of dual simplex iterations with these xings we do

iterations The variable that yields the largest increase in the smaller of the corresp onding

twolower b ound values is the branching variable Our primal heuristic is a plunging metho d

that iteratively rounds fractional variables to the nearest integer and reoptimizes the linear

all variables with values ab ove or if no such variable exists program we round to

the one with the largest value breaking ties arbitrarily This heuristic is called once after

the solution of the initial LP relaxation and once at each no de of the searchtree The default

Anonymous ftp from happygmuedupubacs

An Algorithm for Set Partitioning

strategy for separation is to call the row clique lifting routine the greedy clique detection

the RSF semiheuristic and the GLS cycle algorithm All of these pro cedures are called after

each individual LP Among the violated inequalities that we have found we select the most

violated ones up to a threshold that dep ends on the size of the LP and the number of cuts

found In each iteration cuts with p ositive slack of more than are removed from the

that stops the cutting present LP To avoid tailing o we use an early branching strategy

plane phase if the dualitygap do es not decrease signicantly from one iteration to the next

to within any four successive iterations Like the separation routines the preprocessor

is invoked after each solution of an LP The LPs themselves are solved with the dual simplex

algorithm and steepest edge pricing

We have p erformed three computational experiments to answer the questions iiii Our

Experiment applies BC with the default strategy to the acs test set In Experiment we

also separate aggregated cycle inequalities all other parameter settings are identical For

Experiment we turn o the cut generation mo dule of BC completely ie we apply branch

andb ound with prepro cessing Our results are summarized in Tables

The statistics in these tables have quite some similarities and not only at rst glance We will

in fact argue in our analysis that the outcome of the three exp eriments is essentially the same

except for three hard instances namely nw aaandaa the other problems fall into

a number of categories of readily solvable instances Our discussion will try to explain the

dierences in the computational b ehavior of the instances in terms of two measures of problem

diculty Resp onse to andor size after preprocessing and the initial duality gap Note that

these are a priori criteria ie they are available prior to the solution of the problem and

can be used to predict exp ected solution eorts We remark that we found these indicators

satisfactory not only for the acs problems but also for two sets of Telebus clustering and

chaining instances of dierentcharacteristics from a vehicle scheduling application confer

Section for a discussion of computational results for these instances

A rst similarity is that the initial preprocessing do es not dep end on the dierent parameter

settings of the exp eriments ie the reductions are always the same see the Prepro cessed

columns in Tables We have already given more detailed statistics on the initial

prepro cessing step in Table Taking another lo ok at this data we see that the rst

instances up to nw are reduced to very small problems with less than rows all of

these simple instances can b e solved in well under a second with all strategies

Many of the remaining problems are also fairly small andor display minimal initial duality

vo cation of the primal heuristic and without adding any cutting gaps already after the rst in

planes see column Gap in Table In fact all but of the instances have a duality

gap of or less One would hop e that the solutions of the initial LP relaxations of these

problems are close to integrality ie they have only few fractional variables one can not see

this from the tables and this is indeed the case of the instances haveintegral LP

solutions the remaining fractional solutions are rounded to optimal ones at the ro ot no de

in all but cases by BCs simple plunging heuristic this data is also not in the tables It

is thus not surprising that those of instances with gap can be solved in

ab out seconds with all strategies Note that the solution statistics for the hardest of

these problems instances aa kl aaandaa see the BranchandCut columns

in Table t with our diculty indicators in terms of size and gap The diculty of the

three aa instances is due to a large number of rows which leads to large bases and a relatively

large number of pivots in the LP solution pro cess see column Itn in Table while kl

displays the largest initial duality gap of see column Gap in Table

Computational Results

The remaining nine instances nw nw nw nw kl us nw aaandaa are

the ones that require the use of cutting planes see column Cut in Table several LPs

see column LP and some branchandb ound see column BB The rst ve of these

can again b e solved fast in ab out CPU seconds no matter if many or no cutting planes at

all are used This b ehavior is due to the fast decrease of the duality gap in the ro ot section

of the searchtree In Exp eriment eg the optimum is not found in the rst rounding of

the solution of the initial LP relaxation but it comes up rapidly in trial and

resp ectively recall that the plunging heuristic is called once after the solution of the initial

LP and once at each no de ie a means that the optimum is found rounding the second

LP at the ro ot no de while refers to the rst LP at no de number Comparing these

numbers with the size of the searchtree in column BB reveals that the problems were

solved immediately after this happ ened

The analysis of the previous paragraph applies also to problem us The optimum is found

at the ro ot no de with the second call to the heuristic and then the problem is essentially

nished in all three exp eriments us is not a hard problem but a large one accounting

for ab out of both nonzeros and columns of the entire test set and it just takes some

minutes to pro cess all this data The initial LP alone takes ab out minutes

We are thus indeed left with only three instances where the dierent use of cutting planes

in our exp eriments can make a dierence nw aa and aa Note that these problems

account for out of a total of branchandb ound no des in Exp eriment similar

statements hold for the other exp eriments for out of LPs for out of

dual simplex iterations for out of in and outpivots and for

out of CPU seconds ie the p erformance of our algorithm BC on these four problems

determines the outcome of our computational exp eriments completely We would however

like to stress that the hitherto treated simple instances are formulations of real world

problems and that the ability to solve airline crew scheduling problems to proven optimality

in such short times is one of the most remarkable successes in op erations research To put it

in a p ointed way It is the computational wellb ehaviour that makes set partitioning mo dels

so useful As even the hard problems in the acs test set can b e solved in ab out minutes with

the default strategywe answer question i ab out the p erformance of BC on the acs problems

with a conrmation of Homan Padb erg s conclusion that it is p ossible to solvevery

large setpartitioning problems to proven optimality and that by using the branchandcut

technology describ ed ab ove and solving larger setpartitioning problems exactly than is

done to day the airline industry could see immediate and substantial dollar savings in their

crew costs

The three hard instances themselves fall again into two dierent categories namely instance

nw on the one and aa and aa on the other hand The dierence b etween them is that

nw has few rows and many columns while the aa problems have the opp osite prop erty We

will give now a number of heuristic arguments that suggest that set partitioning problems

with many rows tend to b e more dicult for a branchandcut algorithm than problems with

many columns In fact there are only two o ccasions where BC examines the complete set of

columns In the pricing step of the dual simplex algorithm and in the prepro cessing But these

steps take linear or loglinear time only The more exp ensive mo dules work on data structures

whose size dep ends on the number m of rows Refactorization works on a matrix of size O m

and has quadratic running time separation works on a fractional intersection graph of the

same size and has at least the same order of running time and we exp ect the primal heuristic

to p erform O m rounding steps requiring the same numb er of LP reoptimizations

An Algorithm for Set Partitioning

In light of these arguments it is not surprising that nw can be solved ve to six times

faster than aa and aa In fact the solution time for nw is at most seconds with

any strategy such that one could even question the classication of nw as a hard instance

Lo oking at the solution statistics in the BranchandCut columns of Tables

however shows that nw has the same complexityasthe aa problems Its solution requires

a number of no des LPs simplex iterations and cutting planes that is in the same order of

magnitude as the gures for the aa problems The three hard problems have in common that

the initial LP solution do es not immediately reveal the optimum nor a pro of of optimality

and that the solution takes some algorithmic eort The smaller running time for nw is

solely due to the smaller amount of computation at the individual no des of the searchtree

Recalling how the simple instances nw nw nw nw kl and us could be

solved easily once the optimal solution was found one mightwonder if the hard problems are

dicult b ecause BCs simple plunging heuristic is unable to nd go o d solutions To answer

this question we have run our co de with the optimal solution as an additional input It

turns out that primal knowledge do es not make the problems much easier For the default

strategy eg we still needed no des LPs dual

simplex iterations cuts and CPU seconds to solve

nwaaaa resp ectively the decrease in the running times of nw and aa is mainly

due to a more eective reduced cost xing while aa takes in fact even longer to solve

Closing the gap from the dual side thus seems to be what makes instances nw aa and

aa hard Here is where cutting planes come into play and where the dierent separation

strategies in Exp eriments make a dierence We rst turn to question ii ab out the

signicance of cutting planes for the solution pro cess Comparing the results of Exp eriment

in Table with the default strategy to the outcome of the branchandb ound Exp eriment

in Table gives the disapp ointing result that the negligence of the cuts is not punished

with an increase in running time There is only a redistribution away from cut generation

and the LP to the other mo dules of BC Hence our timing statistics give no arguments in

favor of cutting planes The BranchandCut parts of Tables and however provide

some justication for the use of cutting planes Cuts reduce the size of the searchtree from

no des in Exp eriment to only in Exp eriment and similar alb eit

smaller reductions apply to the numb er of LPs dual simplex iterations and in and outpivots

These ndings do certainly not sp eak against the use of cutting planes in computational set

partitioning

Exp eriment was designed to investigate another step in this direction Do the aggregated

cycle inequalities of Section yield a computational advantage The answer to question iii

is similar to our ndings for question ii Comparing the results of Exp eriment in Table

with the statistics on Exp eriment in Table displays an increase in running time by a

factor of three when aggregated cycle inequalities are used This outcome is however solely

due to the exp erimental status of our aggregated cycle separation routine An examination

of the Cut column in the Timings section of Table shows that ab out of the

running time is sp ent in this mo dule The BranchandCut statistics show some encouraging

eects of aggregated cycle separation The searchtrees are reduced from no des to

no des and similar savings can b e observed for the numb er of LPs and dual simplex

iterations We feel that these results indicate some p otential for aggregated cycle inequalities

and strongly b elieve that cuts of such aggregation typ es are valuable for solving hard integer

programming problems not only set partitioning problems The separation mo dule itself

leaves ample ro om for improvement and this is one of the issues of future research

Computational Results Name n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nn nn aa aa aa aa aa aa us us us us kl kl w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w Ro ws rgnlProblem Original Cols T NNEs be able Ro ws ovn e P Set Solving rpocessed Prepro Cols NNEs riinn rbesb Problems artitioning Solutions z Gap Pvt Branc y Cut Branc handCut handCut Itn LP eal Strategy Default BB PP Pvt Timings Cut LP Heu TimeSec T otal

An Algorithm for Set Partitioning Name n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nn nn aa aa aa aa aa aa us us us us kl kl w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w Ro ws T rgnlProblem Original be able Cols ovn e P Set Solving NNEs Ro ws riinn rbesb Problems artitioning rpocessed Prepro Cols NNEs Solutions z yBranc Gap handCut Pvt Cut Branc eaaigAggregated Separating handCut Itn LP BB PP yl Inequalities Cycle Pvt Timings Cut LP Heu TimeSec T otal

Computational Results Name n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nn nn aa aa aa aa aa aa us us us us kl kl w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w Ro ws rgnlProblem Original Cols NNEs T Ro be able ws rpocessed Prepro Cols ovn e P Set Solving NNEs riinn rbesb Problems artitioning Solutions z Gap Pvt Cut Branc yBranc handCut Itn handBound LP BB PP Pvt Timings Cut LP Heu TimeSec T otal

An Algorithm for Set Partitioning

Bibliography of Part

Ahuja Magnanti Orlin Network Flows In Nemhauser Rinno oy Kan Todd

chapter IV pp

Andersen Andersen Presolving in Linear Programming Math Prog

Applegate BixbyChvatal Co ok Large Scale Combinatorial Optimization Talk at

the th Int Symp on Math Prog Ann Arb or MI

Atamturk Nemhauser Savelsb ergh ACombined Lagrangian Linear Programming

and Implication Heuristic for LargeScale Set Partitioning Problems Tech Rep LEC

Georgia Inst of Tech

Balas Ceria Cornuejols A LiftandPro ject Cutting Plane Algorithm for Mixed

Programs Math Prog

Balas Padb erg Set Partitioning A Survey SIAM Rev

Balinski Integer Programming Metho ds Uses Computation Mgmt Sci

Beasley An Algorithm for Set Covering Problem Europ J on OR

Bellman Hall Eds Combinatorial Analysis Pro c of Symp osia in Applied Math

ematics Providence RI

Bixby Lecture Notes for CAAM Combinatorial Optimization and Integer Pro

gramming Spring Rice Univ Houston TX

Brearley Mitra Williams Analysis of Mathemtical Programming Problems Prior

to Applying the Simplex Metho d Math Prog

Caprara Fischetti BranchandCut Algorithms In DellAmico Maoli Martello

chapter pp

Carraghan Pardalos An exact Algorithm for the Maximum Weight Clique Problem

Operations Research Letters

Chu Beasley Constraint Handling in Genetic Algorithms The Set Partitioning

Problem Working pap er Imp erial College London UK

CPLEX Using the CPLEX Cal lable Library Suite Taho e Blvd Bldg

Incline Village NV USA CPLEX Optimization Inc

CPLEX Using the CPLEX Cal lable Library Alder Avenue Suite In

cline Village NV USA ILOG CPLEX Division Information available at URL

httpwwwcplexcom

Crowder Demb o Mulvey On Rep orting Computational Exp eriments with Mathe

matical Software ACM Transactions on Math Software

Avail at URL httpmscmgamsicacukjebj ebh tml

Inf avail at URL httpwwwcplexcom

BIBLIOGRAPHY OF PART

Crowder Johnson Padb erg Solving LargeScale ZeroOne Linear Programming

Problems Op Res

DellAmico Maoli Martello Eds Annotated Bibliographies in Combinatorial

Optimization John Wiley Sons Ltd Chichester

EmdenWeinert Hougardy Kreuter Pro emel Steger Einfuhrung in Graphen und

Algorithmen

Garey Johnson Computers and Intractability A Guide to the Theory of NP

Completeness New York W H Freeman and Company

Garnkel Nemhauser The Set Partitioning Problem Set Covering with Equality

Constraints Op Res

Gomory Solving Linear Programming Problems in Integers In Bellman Hall

Grotschel Lovasz Schrijver Geometric Algorithms and Combinatorial Optimization

Springer Verlag Berlin

Homan Padb erg Improving LPRepresentations of ZeroOne Linear Programs for

BranchandCut ORSA J on Comp

Homan Padb erg Solving Airline CrewScheduling Problems by BranchAndCut

Mgmt Sci

Jackson Boggs Nash Powell Guidelines for Rep orting Results of Computational

Exp eriments Rep ort of the Ad Ho c Committee Math Prog

Martin Weismantel The Intersection of Knapsack Polyhedra and Extensions

Preprint SC KonradZuseZentrum Berlin

Nemhauser Rinno oy Kan Todd Eds Optimization volume of Handbo oks in

OR and Mgmt Sci Elsevier Sci BV Amsterdam

Nemhauser Wolsey Integer and Combinatorial Optimization John Wiley Sons

Inc

Nobili Sassano A Separation Routine for the Set Covering Polytop e pp

Padb erg Rinaldi A Branch and Cut Algorithm for the Resolution of LargeScale

Symmetric Traveling Salesman Problems SIAM Review

Suhl Szymanski Sup erno de Pro cessing of MixedInteger Mo dels Computational

Optimization and Applications

Thienel ABACUS A BranchAndCUt System PhD thesis Univ zu Koln

Wedelin An algorithm for large scale integer programming with application to

airline crew scheduling Ann Oper Res

Avail at URL httpwwwinformatikhuberlin de Insti tut stru ktur algo rith meng a

Avail at URL httpwwwzibdeZIBbibPublic atio ns

Index of Part

cutting planes A

aggregated cycle inequality acs see airline crew scheduling problems

clique inequality

advanced LP basis

cycle inequality

in a cutting plane algorithm

cycle inequality

inpivoting technique

lifting

outpivoting technique

separation

aggregated cycle inequality

data structures

for the set packing p olytop e

dominated column reduction

lifting

dominated row reduction

separation with the GLS algorithm

dual feasibilitysee advanced LP basis

aggregated fractional intersection graph

duplicate column reduction

for a set covering problem

duplicate row reduction

airline crew scheduling problems

early branching

computational results

empty column reduction

Homan Padb erg test set

emptyrow reduction

preprocessing

ll in column singleton reduction

staircase form

owchart

fractional intersection graph

basissee advanced LP basis

GLS algorithm

BC set partitioning solver

for aggregated cycle separation

advanced LPbasis

for cycle inequality separation

aggregated cycle inequality

greedy heuristic

lifting

for clique inequality separation

separation

inpivoting technique

aggregated fractional intersectn graph

intersection graph

lifting

basissee advanced LP basis

of aggregated cycle inequalities

b est rst search

of clique inequalities

branching strategy

of cycle inequalities

cache usage

LPbasedreduction

clique inequality

no de selection strategy

lifting

outpivoting technique

separation

parallel column reduction

columnmajor format

pivoting

column singleton reduction

inpivoting technique

computational results

outpivoting technique

prepro cessing acs problems

plunging heuristic

p o ol management solving acs problems

INDEX OF PART

C preprocessing

he usage in prepro cessing cac

advanced LP basis

clique inequality

cache usage

for the set packing polytope

column singleton reduction

lifting

computational results

separation

dominated column reductn

with the greedy heuristic

dominated row reduction

with the row lifting heuristic

duplicate column reduction

with the RSF semiheuristic

duplicate row reduction

column intersection graph see intersection

empty column reduction

graph

emptyrow reduction

column ma jor format of a matrix

LPbasedreduction

column singleton reduction

parallel column reduction

complexity

pivoting

of the set covering problem

probing

of the set packingproblem

reduced cost xing

of the set partitioning problem

row cliquereduction

computational results

row singleton reduction

prepro cessing acs problems

symm dierence reductn

solving acs problems

preprocessor

constraint classicationforIPs

primal heuristic

CREW OPT set partitioning solver

probing

cutting plane algorithm

recursive smallest rst semiheuristic

cutting planes in BC

for clique inequality separation

aggregated cycle inequality

reduced cost xing

clique inequality

reductions see prepro cessing

cycle inequality

row clique reduction cycle inequality

for the set packing polytope

row lifting heuristic

lifting

for clique inequality separation

separation with the GLS algorithm

row ma jor format

row singleton reduction

searchtree management

data structures in BC

separation

column ma jor format

of aggregated cycle inequalities

row ma jor format

of clique inequalities

degeneracy see primal degeneracy

of cycle inequalities

diameter of a graph

steep est edge pricing

dominated column reduction

strong branching

dominated row reduction

symmetric dierence reductn

doubleton row inanIP

b est rst search no de selection strategy

dual feasibility see advanced LP basis

branchandcutalgorithm

duplicate column reduction

branching strategies in BC

duplicate row reduction

early branching

strong branching

bubblesort algorithm early branching strategy in BC

INDEX OF PART

intersection graph empty column reduction

aggregated fractional intersectn graph emptyrow reduction

for a set covering problem eta le technique for probing

for a set packing problem exp ected running time

fractional intersection graph for a lexicographic comparison of two

intersection of random sequences see random sequences

sequence in for the sequence intersection algorithm tersection

iterated prepro cessing

of intersection graph construction

of prepro cessing reductions

lexicographic comparison

of the column singleton reduction

of two random sequences

of the dominated row reduction

liftandpro ject cut

of the duplicate column reduction

lifting

of the duplicate row reduction

of aggregated cycle inequalities

of the empty column reduction

of clique inequalities

of the emptyrow reduction

of cycle inequalities

of theGLScycle separation

LP based reductions for IPs

of the greedy clique separation

of the row clique heuristic

of the row clique reduction

no de selection strategy in BC

of the row singleton reduction

of the RSF clique separation

of the symmetric dierence reductn

oset in a set partitioning problem

outpivoting technique

feasibleset cut

ll in column singleton reduction

parallel column reduction

xing a variable

pass of a prepro cessor

owchart of BC

plunging heuristic

fractional intersection graph

p o ol management in BC

aggregated fractional intersectn graph

prepro cessing for IPs

for a set covering problem

constraint classication

for a set packing problem

doubleton rows

LPbasedreductions

probing

GLS algorithm

reducedcost xing

for aggregated cycle inequality separa

reduction prepro cessing op eration

tion

redundant constraint elimination

for cycle inequality separation

substitution reduction

Gomory cut

tightening a formulation

greedy heuristic

tighteningbounds

for cycle inequality separation

prepro cessing for set partitioning

advanced LP basis

basissee advanced LP basis

hashing

cache usage

column ma jor format I

column singleton reduction inpivoting technique

INDEX OF PART

primal degeneracy computational results

in a set partitioning problem data structures

primal heuristic in BC dominated column reduction

probabilistic mo del dominated row reduction

for a lexicographic comparison of two dual feasibilitysee advanced LP basis

random sequences duplicate column reduction

for the sequence intersection algorithm duplicate row reduction

empty column reduction

probing emptyrow reduction

ll in column singleton reduction

inpivoting technique

quicksort algorithm

iterated prepro cessing

lexicographic comparison

of two random sequences

recursive smallest rst semiheuristic

LP based reduction

for clique inequality separation

operations

reduced cost xing

outpivoting technique

reduction prepro cessing op eration

parallel column reduction

redundant constraint elimination for IPs

pass of a prepro cessor

row clique reduction

probabilistic mo del

row lifting heuristic

for lexicographic comparison

for clique inequality separation

for sequence intersection

row major formatof a matrix

probing

row singleton reduction

reduced cost xing

RSF see recursive smallest rst

reductions

S column singleton

searchtree managementinBC dominatedcolumn

semiheuristic dominated row

separation duplicate column

of aggregated cycle inequalities duplicate row

of clique inequalities empty column

with the greedy heuristic emptyrow

with the row lifting heuristic LPbased

with the RSF semiheuristic parallelcolumn

of cycle inequalities probing

sequence intersection algorithm reduced cost xing

set covering p olytop e row clique

aggregated cycle inequality row singleton

set covering problem symmetric dierence

complexity row clique reduction

set packing p olytop e row ma jor format

clique inequality row singleton reduction

cycle inequality sequence intersection algorithm

set pac symmetric dierence reductn king problem

tightening an IP formulation complexity

prepro cessor of BC intersection graph

presolvingsee prepro cessing set partitioningpolytope

INDEX OF PART

set partitioning problem

complexity

oset in the ob jective

preprocessing

primal degeneracy

set covering relaxation

set packing relaxation

staircase form

shakersort algorithm

staircase form

of a set partitioning problem

steep est edge pricing

strong branching strategy in BC

substitution reduction for IPs

symmetric dierence reduction

tighteninganIPformulation

tighteningboundsof anIP

INDEX OF PART

Part III

Application Asp ects

12 I

34 II III 567

89 10 11 IV V

Chapter

Vehicle Scheduling at Telebus

Summary This chapter is ab out set partitioning metho ds for vehicle scheduling in diala

ride systems We consider the optimization of Berlins Telebus for handicapp ed p eople that

services requests per day with a eet of mini buses Our scheduling system is in

use since June and resulted in improved service and signicant cost savings

Acknowledgement The results of this chapter are jointwork with Fridolin Klostermeier

and Christian Kuttner

Co op eration The Telebus pro ject was done at the KonradZuseZentrum Berlin ZIB in

co op eration with the Berliner Zentralausschu fur Soziale Aufgab en eV BZA the Berliner

Senatsverwaltung fur Soziales SenSoz and the Intranetz Gesellschaft fur Informationslo

gistik mbH

Supp ort The Telebus pro ject was supp orted from July to December by

the Berliner Senatsverwaltung fur Wissenschaft Forschung und Kultur SenWiFoKult

Intro duction

This chapter is ab out set partitioning metho ds for vehicle scheduling in dialaride systems

and their application at Berlins Telebus for handicapp ed p eople

Intranetz GmbH Bergstr Berlin GermanyURL httpwwwintranetzde

KonradZuseZentrum Berlin Takustr Berlin GermanyURL httpwwwzibde

Berliner Zentralausschu fur Soziale Aufgab en eV Esplanade Berlin Germany

Senatsverwaltung fur Soziales An der Urania Abt IX B Berlin Germany

Senatsverwaltung fur Wissenschaft Forschung Kultur Abt I I I B Brunnenstr Berlin Germany

Vehicle Scheduling at Telebus

Dialaride systems give rise to challenging optimization problems that involve strategic as

well as op erational planning uncertainty and online asp ects decisions in space and time

complicated feasibility constraints and multiple ob jectives soft data fuzzy rules and

applications of large scale This colorful manifold of topics is matched by the wide variety

of metho ds that have b een develop ed to solve the planning questions of this area Dynamic

programming algorithms network mo dels set partitioningset covering and other integer

programming approaches and all kinds of combinatorial heuristics single and multiphased

clusterrst schedulesecond and vice versa etc For surveys on dialaride problems and

solution metho ds we refer the reader to Desrosiers Dumas Solomon Soumis and

the thesis of Sol Chapter see also Hamer for a recent description of a mo dern

largescale dialaride system for handicapp ed p eople in Toronto and the thesis of Tesch

German for the example of the German cityofPassau

We discuss in this chapter the application of some of these optimization metho ds to vehicle

scheduling in a sp ecic dialaride system Berlins Telebus for handicapp ed p eople Our

approach is based on a decomposition of this dialaride problem into a clustering and a

chaining step Both of these steps lead to set partitioning problems that we attack with

heuristic and branchandcut metho ds These pro cedures form an optimization mo dule that

is the heart of a computer system that integrates and automates the complete op eration of

the Telebus center This system is in use since June and lead to improvements in

service cost reductions and increased pro ductivity of the center

This chapter is organized in seven sections in addition to this intro duction Section de

scrib es Berlins Telebus transp ortation system for handicapp ed p eople and our pro ject to

optimize the op eration of this service The core of the pro ject was the development of mathe

matical optimization metho ds and software to solvethevehicle scheduling problem that comes

up at Telebus this particular dialaride problem is discussed in Section Section in

tro duces our twophase clustering and chaining solution approach and the asso ciated set par

titioning mo dels The approachinvolves cluster and tour generation steps that are discussed

in Sections and Computational exp eriences with our vehicle scheduling metho d are

discussed in Section and some p ossible future p ersp ectives in the nal Section

Figure ATelebus Picks Up a Customer

Telebus

Accessibility of the public transp ortation system for mobility disabled people has b ecome an

imp ortant p olitical goal for manymunicipalities They intro duce lowo or buses install lifts

in subway stations etc But many handicapp ed and elderly p eople still have problems b ecause

they need additional help the next station is to o far away or the line they want to use is not

yet accessible Berlin like many other cities oers to these p eople a sp ecial transp ortation

service The socalled Telebus provides i doortodoor transportation and ii assistance at

the pickup and the dropo point The system is op erated by the Berliner Zentralausschu

fur Soziale Aufgab en eV BZA an asso ciation of charitable organizations and nanced

by the Berliner Senatsverwaltung fur Soziales SenSoz the city of Berlins Department for

So cial Aairs Figure on page shows a Telebus that picks up a customer

Telebus is a dialaride system Every entitled user currently ab out p eople can order

up to rides per month through the BZAs telephone center If the order is placed one

day in advance Telebus guarantees to service the ride as requested later sp ontaneous

requests are serviced as p ossible The advance orders ab out during the week and

on weekends are collected and scheduled into a eet of minibuses that T elebus rents

on demand from service providers like charitable organizations and commercial companies

These buses pick up the customers at the requested time mo dulo a certain tolerance and

transp ort himher to the desired destination if required the crew provides assistance to leave

the apartment enter the vehicle etc This service is available every day from am in the

morning to am in the night Figures and illustrate op eration and organization of

the Telebus system

order customer

telephone center BZA

bus scheduling BZA

bus renting BZA

transp ortation service provider

nancing goals SenSoz

Figure Op eration of the Telebus System

Telebus was established in and ever since then the numb er of customers and requests has

b een rapidly increasing Figure on page gives an impression of the dramatic history of

Telebus in this p erio d of time the numb ers for the years up to are taken from T of

the rep ort of the Rechnungshof von Berlin Berlins audit division for the year the other

data was provided by the BZA We see rst that there is a constant growth in the number of

entitled users But not all registered p ersons drive The number of customers ie p ersons

that order rides was basically constant and started to increase only after the reunication

of Germany in the delay until is due to the initial lack of private telephones in

the East Costs got out of control in when a taxi voucher system was intro duced that

allowed for a certain number of sp ontaneous rides with taxis in addition to the bus rides

Vehicle Scheduling at Telebus

Figure Organization of the Telebus System

Telebus

When costs topp ed million DM in drastic service reductions were taken to stop this

trend The voucher system was replaced by a taxi account system that limits taxi rides to

DM p er p erson and month But with new demand from East Berlin and a doubled area

of service costs were almost immediately up at million DM again What could b e done

Telebus Computer System

Taxi AccountSystem

Taxi Voucher System

Year

Costs in Million DM

Entitled Users in Thousands

Customers in Thousands

Figure Increasing Usage and Costs of Telebus

The b est waytocontrol costs without reducing the service was a b etter vehicle scheduling to

service more requests for the same amountofmoney The scheduling was traditionally done

manual ly by exp erienced planners who could work out a feasible bus plot in ab out man

hours Now it b ecame clear that this metho d was no longer appropriate to cop e with rising

demand and cost pressure The core scheduling problem of the BZA could only be solved

with mo dern computer hard and software and the Telebus project a co op eration involving

the ZIB the BZA and the SenSoz Intranetz joined later see next paragraph was started

to develop it The Telebus dialaride problem the metho ds that we use for its solution and

t sections of our computational exp eriences are what we are going to discuss in the subsequen

this chapter

The pro ject develop ed a broader scop e It so on turned out that a mathematical vehicle

scheduling to ol alone was not enough and the pro ject evolved quickly into the development

of a comprehensive Telebus computer system that integrates and automates the complete

op eration of the BZA Reservation conrmation and cancellation vehicle scheduling radio

telephony accounting controlling and statistics The system consists of a to ol b ox of software

mo dules and runs on a network of MacIntosh PCs it is in op eration since June

Design and installation of the Telebus computer system lead further to a reorganization of

the center and the whole Telebus service with issues that ranged from a new bus renting

Vehicle Scheduling at Telebus

mix and mo de to scheduling training of BZA sta Fridolin Klostermeier and Christian

Kuttner in particular worked with great p ersonal dedication for more than a year in the

Telebus center drove on Telebuses etc Finally they even set up their own company the

Intranetz GmbH that has scheduling systems for dialaride systems as one of its business

areas More information on the consulting aspect of the Telebus pro ject can b e found in the

articles Borndorfer et al ab and the thesis of Klostermeier Kuttner all

these publications are in German

All these measures together negotiations with vehicle providers reorganization of center

and service the new computer system and improved vehicle scheduling resulted in

ements in service A reduction of the advance cal lin period p erio d from three i Improv

days in to one day and increased punctuality of the schedule in comparison to the

results of manual planning

ii Cost reductions Today ab out more requests can be serviced with the same

resources as in see Figure for a comparison of a month in b efore and in

after the Telebus computer system wentinto op eration

iii More pro ductivityintheTelebus center

Requests Costs in DM 40.000 2.000.000

30.000 1.500.000 20.000 1.000.000 10.000 500.000

0 0 Mai

Mai

Figure Results of the Telebus Pro ject

The Vehicle Scheduling Problem

The most imp ortant task at Telebus is the daily construction of the vehicle schedule which

determines the two most imp ortant ob jectives of the service Operation costs and customer

satisfaction This vehicle scheduling problem is a dialaride problem that can be stated in

an informal wayasfollows

Given the customer requests rent a suitable set of available vehicles and schedule

DARP all requests into them such that a number of constraints on the feasibility of

vehicle tours are satised and op eration costs are minimum

In the remainder of this section we discuss the Telebus D ARP in detail

The Vehicle Scheduling Problem

Requests

day

Bus Bus Taxi Cancelled

Figure Telebus Requests in June

Pieces of Work

The basis for vehicle scheduling are the vehicles Vehicles always come together with a crew

for a p ossible shift of op eration following the terminology of Desrosiers Dumas Soumis

we call suchapartorallofaworkday during which a crew and a vehicle is available

to service requests a piece of work The supply of pieces of work is determined by the vehicle

providers who oer ab out pieces of work of dierenttyp es The available pieces of work

are known in advance and Telebus can rent them on a dailyweekly or monthly basis long

term renting can b e cheap er

The following data is asso ciated to a piece of work w

i v vehicle typ e Teletaxi busbus smalllarge

c w nf f a

ii cv c c c c c v capacity

w w

total of customers and wheelchairs

of nonfoldingfolding wheelchairs

ofambulatories

iii Gw group vehicle typ e dep ot typ e of shift

The vehicle types are ve Teletaxis taxis that are rented like buses buses with one

driver that can be small or large and buses also small or large The vehicle typ e is

imp ortant for deciding whether a request can be serviced by a particular piece of work

Teletaxis can transp ort only ambulatories and customers with folding wheelchairs nonfolding

wheelchairs require a bus and staircase assistance a bus see Figure for statistics on

Telebus requests whichshowa typical weekly distribution pattern

Each vehicle has a capacity that dep ends on the typ e It can transp ort c v p ersons in

nf w

folding and c v p ersons in nonfolding wheelchairs but at most c v wheelchairs at

w w

a c

the same time plus c v ambulatories in total at most c v customers Teletaxis have

w w

a capacity of cv ie they can service up to three customers at the same

time and one of them can have a folding wheelchair The small buses have a capacity of

cv large buses have cv note that this allows to account

w w

for the Telebus rule that p ersons in folding wheelchairs travel in buses in their wheelchair

Vehicle Scheduling at Telebus

G A group G is Finally the set W of all pieces of work falls into disjoint groups W

a set of pieces of work that are considered to b e indistinguishable for the purp ose of vehicle

scheduling ie the pieces of work of the same group havevehicles of the same typ e which are

stationed at the same depot and they can b e rented for identical shifts of op eration p ossible

shifts are and hours xed length and early late and certain exible shifts of variable

duration The groups will b ecome imp ortant for the construction of vehicle tours namely

we will require group sp ecic parameter settings or even group sp ecic algorithms to come

up with tours that can b e serviced by the pieces of a work of a given group

Requests

The pieces of work will b e used to service some number m of transp ortation re quests

i v v pickup and dropo event

i i

ii pv pv pickup and dropo lo cation

i i

t pickup time window iii T v v tv

i i i

t T v dropo time window v tv

i i i

iv t v t v pickup and dropo service time R

i i

v W v Wv set of feasible pieces of work

i i

c w nf f a

vi av a a a a a v total of customers and wheelchairs

i i

c w nf f a

av a a a a a v of nonfolding and folding wheelchairs

i i

ofambulatories

opo node v that corresp onds to Asso ciated to each request i is a pickup node v and a dr

i i

the pickup and dropo event of a request these no des will b e part of a spacetime transition

network that will b e dened in the next section

The pickup and the dropo no des are mapp ed to locations or p oints pv andpv in a

i i

road network of Berlin dierent from the transition network that is shown in Figure We

estimate travel ling times and distances byaverage values that are stored on the edges

of this network and use this data to compute shortest routes b etween the no des

In addition to this spatial information a request b ears temp oral data that is measured in

units of minutes of Telebus time The customer communicates a desired pickup time t v

or a desired dropo time whic h is treated analogously that gives rise to a window of feasible

The pickup time window is computed according to Telebus sp ecic pickup times T v

rules that try to nd a compromise between punctual service and more degrees of freedom

for the vehicle scheduling pro cess Currently most requests have

T v t v units of Telebus time

i i

ie the vehicle is allowed to arrive up to minutes early or late Similar but more complex

Here the shortest p ossible rules are used to determine a feasible dropo time window T v

travelling time and a maximum detour time play a role Finally some service time t v

is needed at the pickup and the dropo lo cation and t v

The required assistance the wheelchair typ e etc determine the set of feasible pieces of

work W v W v that can or must be used to service the request this set consists

i i

of all suitable groups

av and av give the total number of customers and wheelchairs the numb er of folding

i i

and nonfolding wheelchairs and the number of ambulatories that enter and leave the vehicle

in the pickup and dropo event resp ectively

The Vehicle Scheduling Problem

Figure Highways and Ma jor Roads in Berlin

Constraints

Given the available pieces of work and the requests a schedule of feasible vehicle tours has to

b e determined that satises a number of constraints Following Desrosiers Dumas Soumis

we distinguish the following typ es of constraints for feasibility

i visiting each pickup and dropo event has to b e serviced exactly once

ii pairing pickup and dropo of a request is serviced by the same vehicle

iii precedence each customer must b e picked up b efore dropp ed o

iv time window pickup and dropo events must b e serviced in time

F v no stop it is not allowed to stop and wait with a customer on b oard

vi capacity the vehicle capacitymust not b e exceeded

vii depot the vehicle must return to its dep ot

viii shift each piece of work must conform to its typ e of shift

ix availability one can not use more pieces of work or others than available

Shift constraints arise from renting contracts and lab our regulations for bus drivers At

Telebus pieces of work have to b e of certain xed or maximum lengths andor have to b egin

and end in certain time intervals the exact parameters dep end on the typ e of shift Suchtyp es

are for example or hour shifts b etween am and am and exible shifts of

variable length Lab our regulations prescrib e maximum driving hours and obligatory breaks

A break of minutes has to b e taken b etween the fourth and sixth hour of a shift

The meaning of the other constraints is self explanatory

Ob jectives

The main objective of the DARP is to minimize operation costs ie the costs for renting

pieces of work from the service pro viders Customer satisfaction is another imp ortant goal

it is treated by means of the time windows Finally Telebus uses some auxiliary ob jectives

that reect security issues These criteria try to prefer safe tours to risky or packed

ones in an attempt to safeguard against emergency situations and unpredictable events like

cancellations sp ontaneous requests trac jams vehicle breakdowns etc

Vehicle Scheduling at Telebus

Solution Approach

In this section we discuss our solution approach to the Telebus dialaride problem Starting

from a network formulation of the DARPwe decomp ose the problem into a clustering and a

chaining step Both steps lead to set partitioning problems

Transition Network

The basis for our solution approach is a formulation of the DARP in terms of a transition

network D V A The transition network is a spacetime digraph see Figure for an

illustration of the following construction

Space

Time

Dep ot Request Request Request Dep ot

Time

Figure Constructing a Transition Network

G G

The transition networks set of nodes V V V V V V consists of all pickup

events V fv g all dropo events V fv g tour start nodes V fv g and

i i

tour end nodes V fv g for each group G of pieces of work and each p ossible p ointof

Telebus time t min utes is am and minutes is am on the

g for all Telebus times Weset next day and break nodes V fv

G G

T v T v T v ftg

t t

G G

and t v t v t v

t t

G G

av av av and

t t

G G

W v G and W v W W v

t t

The arcs of the network are dened to reect the lo cal feasibility of p ossible vehicle tours

Wedrawanevent arc uv between twoeventnodesu and v if

t ut ut tv

that is if it is p ossible to arrive at u service u drive to v and arrive there in time Here

we denote by t the shortest time to get from lo cation pu to pv in the road network

We use here the same symbol t for indices and variables but we hop e the notation is nevertheless clear

Solution Approach

Analogouslyweintro duce tour start arcs from the tour start no des to the event no des tour

end arcs from the event no des to the tour end no des and break start arcs and break end arcs

from the event to the break no des and vice versa More precisely we draw a tour start arc

G G

v v from a tour start no de u v to a pickup no de v if

t t

tv and W v G t t

where t is the time to get from the lo cation of the dep ot asso ciated to the group G of pieces

of work to the event lo cation pv Tour end break start and break end arcs are dened in

the same way only that the break arcs get zero duration t t

vv v v

t t

With this terminology we can state the DARP in terms of the transition network Feasible

vehicle tours corresp ond to such dipaths in D that satisfy the constraints F iiviii as

stated in Subsection on page we assume here that a break is taken by visiting a

break no de A feasible vehicle schedule is a collection of feasible vehicle tours that satises

the remaining constraints F i and ix as well The DARP is the problem to nd a best

p ossible schedule with resp ect to some not yet precisely dened ob jective function

Decomp osition

The construction of feasible vehicle tours in the transition network as dipaths sub ject to

additional constraints is although simple in principle dicult in practice b ecause of the

many constraints F We use a decomposition approach to cop e with this diculty in a

heuristic way The metho d fo cusses on lo cal feasibilityina rst step When validity of the

lo cal constraints is ensured we deal with the remaining global restrictions in a second step

12 I

34 II III 567

89 10 11 IV V

I Collection with Common Dropo

IV Concatenation

II Insertion

V Continued Concatenation

III Collection with Common Pickup

Figure Clusters at Telebus

The decomp osition is based on the concept of a cluster or as schedulers at the BZA say a

Verknupfung A cluster is a dipath in the transition network that qualies as a segment

of a vehicle tour in the sense that it satises the lo cal constraints F iivi Pairing

precedence time windows no stop and capacity Figure shows a number of typical

clusters at Telebus Col lections insertions simple and continued concatenations

Vehicle Scheduling at Telebus

We denote a clusterdipath C v v as a sequence of visited no des In doing so we

want to adopt the convention that a cluster contains only pickup and dropo no des ie no

tour start tour end or break no des We also stipulate that a cluster contains a no de only

once

A cluster C satises the pairing constraints if it contains for every pickup no de the corre

sp onding dropo no de and vice versa The precedence constraints hold if every pickup no de

of the cluster precedes its dropo counterpart Wesay that the capacity constraints are valid

for C if there exists a piece of work w such that the capacity cv of the asso ciated vehicle

typ e is alwa ys at least as large as the load a C ateachnodev of the cluster

cv a C av i k

w i

The time window and the no stop constraints hold if the recursion

T C T v

i i

T v i k T C T C t v t

i i

v v

that computes the feasible time windows at the cluster nodes terminates with T C

Here wedenotebya bt the interval a t b t In this case it is p ossible to start the

service of the cluster at the rst no de v at a feasible time in T C service v goimmediately

no stop to the next no de arrive there at a feasible time T C service v and so on until

the vehicle arrives at the last no de v in the feasible time interval T C

Before we discuss the use of clusters for vehicle scheduling let us record the data that we

asso ciate with a cluster

i C v v cluster as sequence of visited no des

ii t C cluster service time

iii T C cluster start time window

iv W C set of feasible pieces of work

By the no stop constraints and recursion the service time of a cluster is constant

P P

k k

t t v t C

i i

v v

i i

This results in a cluster start time window of p ossible times to b egin the service of a cluster

T C t C tC T C t C

Finally there is the set

W C fw W v j a C cv i kg

i w

of pieces of work that can p ossibly service C

Clusters are useful for vehicle scheduling b ecause they can serve as the building blo cks of

vehicle tours We can chain clusters to feasible tours just as we constructed clusters from

the individual requests As the clusters already satisfy the lo cal constraints F iivi the

conditions F i and viiix the only chaining can concentrate on the remaining global

lo cal constraints that app ear again are the time window constraints F iv that transform

into the cluster start time windows This largely independent treatment of lo cal and global

constraints is one of the main b enets of request clustering

Solution Approach

These observations suggest the following twostepdecomposition approach to the DARP

i Clustering Step Construct a set of feasible clusters

ii Chaining Step Chain clusters to a set of tours that constitute a feasible schedule

This generic clusteringchaining metho d is the vehicle scheduling pro cedure that we use for

the solution of the DARP

Set Partitioning

A renement of the two steps of the clusteringchaining vehicle scheduling metho d leads to

clustering and chaining set partitioning problems of identical structure

The ob jectiveoftheclustering step is to construct a set of clusters that is go o d in the sense

that it provides a reasonable input for the chaining phase In the b est case the clustering

algorithm would yield a set of clusters that can be chained to an optimal solution of the

DARP While this is of course a hop eless criterion one can lo ok for computable necessary

conditions that an optimal set of clusters must satisfy If such conditions can b e found they

can b e used as a measure for the quality of a set of clusters and as a guide to construct it

One way to derivea necessary condition is to note that any feasible schedule decomp oses in

a canonical way The maximal tour segments such that the vehicle is always loaded form a

set of minimal clusters with resp ect to set inclusion interpreting for the moment clusters

as sets of no des and this minimal cluster decomposition of a schedule is unique Then a

necessary condition for the global optimalityofaschedule is that its cluster decomp osition is

also local ly optimal in the sense that the ob jective can not b e improved byrescheduling the

service of individual clusters

Assuming that a lo cal ob jective value can be asso ciated to and computed for an individual

cluster we can approximate the global ob jective value of the schedule by the sum of the

lo cal cluster ob jectives Applying this simplication to the DARP results in the following

optimization problem over clusters

Given the customer requests nd a set of clusters such that each request

CLUSTER

is contained in exactly one cluster and the sum of the cluster ob jectives is

minimal

clustering step this mo del aims at inputs for We use CLUSTER as our formulation of the

the chaining phase that are optimal in a heuristic but wel ldened sense

Popular lo cal optimization criteria for clustering are the internal travel ling time ITT of a

vehicle in a cluster the internal travel ling distance ITD and mixtures of these Clusters

with small ITT or ITD aim at a go o d vehicle usage in terms of transp orted customers per

kilometer or p er minute One would exp ect that a minimal ITT or ITD clustering makes use

of large vehicle capacities by transp orting several customers at once where p ossible see again

Figure for examples in this direction In other words Minimal ITT or ITD clustering

yields reasonable results that planners accept in practice

CLUSTER can b e formulated as a set partitioning problem

T n

SPP min c x Ax x f g

where A is the m n incidence matrix of requests versus clusters and c R is the vector of

cluster ob jectives

Vehicle Scheduling at Telebus

Having decided for a set of clusters we can treat the chaining step in exactly the same way

as we just did with the clustering step Approximating or even expressing the ob jective

value of the DARP as a sum of ob jectives of individual tours the DARP for xed clusters

simplies to b ecomes the following optimization problem over tours

Given a clustering nd a set of vehicle tours suchthateach cluster is contained

CHAIN

in exactly one tour and the sum of the tour ob jectives is minimal

We use CHAIN as our formulation for the chaining step natural ob jectives asso ciated to

tours are op eration costs for vehicles andor customer satisfaction criteria like accumulated

waiting time

CHAIN can also b e mo delled as a set partitioning problem but needs one additional thought

A records the incidences of clusters versus tours and c is the In the simplest case the matrix

vector of tour costs this mo del is correct if there are no availability constraints The presence

of availability constraints leads to additional equations and variables but the enlarged mo del

is again of set partitioning typ e Namely availability constraints for a piece of work w

prescrib e that one can only cho ose at most one of the incidence vectors of tours A that

J w

corresp ond to w

j J w

Adding a slack variable and app ending the new row and column to SPP results again in a

set partitioning problem

A Vehicle Scheduling Algorithm

We are now ready to state the vehicle scheduling algorithm that we prop ose for the solution

of the DARP The algorithm is a renemen t of the generic clusteringchaining metho d of

Subsection in terms of the clustering problem CLUSTER and in terms of a subproblem

of the chaining problem CHAIN

i Cluster Generation Enumerate al l possible feasible clusters Set up its clustering SPP

ii Clustering Solve the clustering SPP

iii Tour Generation Enumerate a subset of all feasible tours Set up its chaining SPP

iv Chaining Solvethechaining SPP

Here the term setting up the clustering SPP for a set of clusters means to construct the

requestcluster incidence matrix A and to compute the cluster cost vector c to set up a

clustering set partitioning problem for the given set of clusters The analogous expressions

the chaining case but here we enumerate only some subset of all feasible tours are used in

to construct only a submatrix of the complete clustertour incidence matrix and a subvector

of the complete tour cost vector The resulting chaining set partitioning problem is hence a

subproblem of the complete chaining SPP The reason for this simplication is that it is out

of the question to set up the complete chaining SPP The numb er of p ossible tours is in the

zillions where zillion is an incredibly large numb er Restricting the chaining SPP to some

subset of promising tours is our heuristic way of dealing with this diculty

We use the branchandcut algorithm BC see Chapter of this thesis to solve the clustering

and chaining set partitioning problems Howwe do the cluster and tour generation is describ ed

in the following Sections and

Cluster Generation

Related Literature

Dialaride problems and solution approaches similar to ours have b een discussed in a number

of publications in the literature In Bo din Golden s vehicle routing and scheduling

classication scheme the Telebus DARP qualies as a subscriber dialaride routing and

scheduling problem and our metho d is a clusterrst schedulesecond algorithm with some of

the clustering transferred to the schedulingchaining phase For survey articles on dialaride

and more general vehicle routing and scheduling problems we give the classic Assad Ball

Bo din Golden see Sections for DARPs the thesis of Sol Chapter

Desrosiers Dumas Solomon Soumis see Chapter for DARPs and the annotated

bibliography of Lap orte and we suggest Barnhart et al and the literature

synopsis of Soumis for references on column generation tec hniques

The termini clustering and chaining stem from Cullen Jarvis Ratli who de

velop a set partitioning based twophase clusteringchaining vehicle routing algorithm Their

approach diers from the one we give here in the use of column generation techniques and

a p ossible overlap of clusters in the chaining phase Ioachim Desrosiers Dumas Solomon

based on earlier work of Desrosiers Dumas Soumis rep ort ab out clustering

algorithms for vehicle routing problems in handicapp ed p eoples transp ort using column gen

eration and a problem decomp osition into time slices Tesch dev elops a set partitioning

metho d that optimizes over a xed set of heuristically generated columns to solve dialaride

problems that come up in the German cityofPassau We nally mention Sol as a recen t

reference for the use of column generation techniques for pickup and delivery problems

Cluster Generation

We discuss in this section the algorithm that we suggest for cluster generation Recursive

enumeration of all dipaths in the transition network D V A that corresp ond to feasible

clusters by depth rst search

The pro cedure works with sequences of no des that are extended in all p ossible ways until

they eventually form clusters Such a sequence S v v V is called a state where

V denotes the set of all nite sequences of elements of V Not every state can b e extended

to a cluster necessary conditions for feasibility of a state S are

i event sequence S V V S contains only eventnodes

i j

ii no loop v v i j S contains each no de at most once

i j

iii precedence v v v v ij pickups precede dropos

p p

iv time windows no stop T S the state time window is nonempty

v capacity availability W S there is a feasible piece of work

Here we use the expressions time window of a state and its set of feasible pieces of work in

analogy to the terms for clusters of the same name see the denitions on page A state

is a cluster or terminal if it is feasible and the constraintfor

i j

v v there is a dropo for each pickup vi pairing v v

p p

k k

holds Finally a new state S v v is pro duced from S v v by app ending

anodev this transition is denoted by

S S v

Vehicle Scheduling at Telebus

To state our depth rst search cluster generation algorithm in Ctyp e pseudo co de we intro

duce predicates infeasible V f g and terminal V f g for feasibility and

terminality of states tail S V is a function that returns the terminal no de v of a state

and the pro cedure output is supp osed to compute the cost and the requeststate incidence

vector of its argument and to store the result somewhere

void dfs state S digraph DVA

f void cluster digraph DVA

if infeasible S return f

if terminal S output S

for all v V

dfs v D

for all u tailS

dfs S u D

Figure Enumerating Clusters by Depth First Search

v The complete cluster generation pro cedure cluster is given in Figure Here

denotes the set of endno des of arcs in D that go out from v

The running time of cluster can b e improved by strengthening the predicate infeasible by

further state elimination criteria For example S is infeasible when it contains an unserviced

pickup v that can not b e dropp ed o in time regardless how S is extended

vii timeout v v S t tv t v can not b e dropp ed in time

p p p

v v

cluster with the infeasible predicate strengthened by vii is the cluster generation

algorithm that we use for vehicle scheduling at Telebus To make this metho d work in

practice one needs of course ecient data structures recursive up dates of the predicates

and many other ingredients the reader can nd the implementation details in the thesis of

Klostermeier Kuttner German For a typical Telebus DARP with requests

our depth rst search pro cedure enumerates the complete set of all p ossible clusters in a

couple of minutes Dep ending on the values of the time window lateness detour and some

other BZA parameters for cluster feasibility this usually results in sometimes up to

feasible clusters for input into the clustering set partitioning mo del

We remark that similar results are not rep orted for comparable clustering problems in the

literature For instance Ioachim Desrosiers Dumas Solomon develop a multilabel

shortest path algorithms for cluster generation problems that come up in the optimization of

Torontos WheelTrans Service Although this dynamic program uses elab orate state space

elimination criteria sp ecial initialization strategies and data structures and sophisticated

not prepro cessing techniques to reduce the size of the transition network it is in this case

p ossible to enumerate all feasible clusters

Two Telebus sp ecic factors may be resp onsible for the dierent outcome in our case One

is the combination of average service driving and detour times As a rule of thumb a

transp ortation service takes in Berlin minutes for pickup minutes of driving and another

minutes for dropo When the maximum detour time is minutes one will already be

happytopick up a single additional customer en route Second not every technically feasible

cluster is accepted by BZA schedulers To safeguards against accumulating delays etc they

often imp ose additional restrictions and forbid continued concatenations ab ove a maximum

length These tw o factors limit the numb er of feasible clusters to a computable quantity

Tour Generation

The topic of this section are tour generation algorithms that chain clusters to feasible vehicle

tours Starting from a simplied network formulation of the chaining problem we develop

a recursive depth rst search tour enumeration algorithm and a number of tour generation

heuristics Some of these heuristics can also b e used as standalone vehicle scheduling to ols

Chaining Network

D The tour generation algorithms of this section work on a chaining transition network

VA that one obtains from the transition network D V Abya contraction of clusters

Togive a more precise description of this construction let C V V b e a clustering of

D is set up from the transition network D in two steps We i delete for each cluster requests

C v v all entering and leaving arcs except the ones that enter the rst no de v and

k k

the ones that leave the last no de v ie we delete all arcs from C n v v

When this has b een done we ii contract each cluster into a single sup erno de that we

denote with the same symbol C Note that we inherit in this w ay the denitions for the

service time of a clusterno de t C its start time interval T C and its set W C of

feasible pieces of work

Tour Enumeration

Feasible vehicle tours corresp ond to dipaths in the chaining network that satisfy the con

straints F Such dipaths can be enumerated in much the same way as the clusterdipaths

in the transition network bya depth rst search pro cedure Using identical terminology and

analogous denitions as for the cluster generation a state S v v V is feasible

when the following conditions hold

i tour start G t v v start at a tour start no de

i j

ii no loop v v i j S contains eachnodeatmostonce

iii time windows T S the state time window is nonempty

iv availability W S there is a feasible piece of work

v shift t t v v v break during thth hour of shift

tv tv tG maximum shift length resp ected

etc

Here we denote by tG the maximum duration of a piece of work of group G The only

dierence to cluster generation is the up date of the time window that must allow for waiting

stops b etween the service of two clusters

R T v i k T S T S t S t

i i

v v

A state is a tour or terminal if it is feasible and the

G G

tour start and end at the same dep ot vi depot G t t v v v v

t t

constraint holds Continuing with the dipath enumeration exactly as we did for the cluster

generation in Section we arrive at a very similar depth rst search tour enumeration

routine chain see Figure on the next page for a Ctyp e pseudo co de listing

Vehicle Scheduling at Telebus

void dfs state S digraph DVA

f void chain digraph DVA

if infeasible S return f

if terminal S output S

for all v V

D dfs v

for all u tailS

dfs S u D

Figure Enumerating Tours by Depth First Search

Computational practice however turns out to b e completely dierent While there were only

some hundred thousand feasible clusters the number of tours is zillions The reason for this

change is not the additional tour start tour end and break no des there are many but not

to o many of these but the p ossibilitytowait b etween the service of two clusters This degree

of freedom that is not available for cluster generation leads to an enormous increase in the

number of eligible clusters to extend a state Lo oking one hour in the future any cluster

qualies as a p ossible followon Unlike in clustering tour state extension do es not have a

lo cal character and although the chain routine worksasfastasthecluster generator there

is no p oint in attempting a complete en umeration of all feasible tours

The way that we deal with this dicultyis by reducing the number of arcs in the chaining

network heuristically One of our strategies is for example to keep only a constantnumber of

outgoing arcs at each cluster no de that are selected by lo cal criteria the k b est successors

While such metho ds are likely to pro duce individual ecient tours in some numb er there is

no reason other than pure luck to b elieve that the right set of unavoidable garbage collection

tours that complete a go o d schedule will also b e pro duced in this way Weareaware of this

fact and mention this unsatisfactory arc selection as a weak point in our vehicle scheduling

algorithm What we can do however is to pro duce in some minutes of computation time

several hundred thousands of tours as input for the chaining set partitioning problem see

again Klostermeier Kuttner for more implemen tation asp ects

Lob el has dealt in his thesis with a similar arc selection problem in the context of

multiple depot vehicle scheduling He has develop ed a Lagrangean pricing technique that

resolves this issue for his extremely large scale problems completely It is p erhaps p ossible

to use this technique based on a suitable multi commodity ow relaxation of the DARP to

obtain b etter chaining results and wehave in fact p erformed some preliminary computational

exp eriments in this direction These computations indicated in our opinion a signicant

p otential for this approach We remark that a multi commo dity ow relaxation gives also

rise to lower bounds for the complete chaining problem CHAIN

Heuristics

The chaining transition network D can also serve as a basis for all kinds of combinatorial

vehicle scheduling heuristics to pro duce individual tours or to pro duce complete schedules

Heuristic scheduling is a particularly attractive metho d of tour generation b ecause it provides

not only reasonable input for the chaining set partitioning problem but also primal solu

tions and upper bounds Wegive here a list of heuristics that wehavedevelop ed for Telebus

a more detailed description can b e found in Klostermeier Kuttner

Tour Generation

Our rst metho d is designed for the construction of individual tours

K Best Neighbors Wehave already mentioned the idea of the k best neighbors heuristic

Applying the depth rst search algorithm chain to a reduced version of the chaining network

where at most k arcs have b een selected from eachset v of arcs that go out from a no de

We use in our implementation lo cal criteria like proximity to select the arcs that lead to a

no des k b est neighb ors

The following heuristics pro duce complete vehicle schedules

TourbyTour Greedy This heuristic pro duces the tours of a complete vehicle schedule

iteratively one by one Starting from some tour start no de the tour is extended by b est

tting followon clusters including breaks in a greedy wayuntil a tour end no de is reached

The serviced clusters are removed from the chaining network the next tour is started and

so on The tourbytour greedy heuristic tends to pro duce go o d tours in the b eginning and

worse later when only farout or otherwise unattractive clusters are left

Time Sweep This metho d uses some linear order on the clusters the time The planning

pro cess constructs all tours of a complete schedule simultaneously In each step of the time

sweep the next cluster with resp ect to the given order is assigned to the b est tour with resp ect

to some lo cal criterion until all clusters are scheduled into tours The orders that we use are

the natural ones from morning to evening and vice versa and a p eaks rst variant that

tries to smo oth the morning and afterno on demand peak byscheduling this demand rst

Hybrid The tourbytour and the time sweep heuristic can b e seen as the extreme represen

tatives of a class of vehicle scheduling heuristics that vary from the construction of individual

tours to a simultaneous construction of all tours by assigning clusters to tours in some order

Hybrid b elongs to such a class of mixtures of these two pro cedures It do es a time sweep but

it adds not only one followon cluster to a tour but some sequence of several clusters

Assignment This metho d b elongs to the same class as the hybrid heuristic but it aims

at some global ov erview The assignment heuristic sub divides the planning horizon into time

slices we use a length of minutes that are considered in the natural order In each step

a b est assignment with resp ect to some lo cal criterion of all clusters in the next time slice

to the current set of partial tours is computed starting new tours if necessary

BZA A set of other metho ds imitates the traditional handplanning metho ds of the BZA

First the request clusters are group ed according to time and space such that the clusters in

one group start in the same hour and city district or similar criteria Doing a time sweep

from morning to evening one constructs tours with an eye on the distribution of clusters and

vehicles in the city districts In the starting phase of the Telebus pro ject these metho ds were

particularly imp ortantto build up condence in computerized scheduling b ecause they can

b e used to pro duce vehicle schedules of familiar typ e

The heuristic vehicle scheduling metho ds that we have just describ ed already pro duce in

a few minutes schedules that have signicantly lower op eration costs than the results of a

manual planning And they do not use a p osteriori changes of scheduling rules that is

they do not pro duce infeasible tours which lead to a quantum leap in punctuality of the

schedule Klostermeier Kuttner give a detailed account of these improvements

We use the heuristic metho ds of this subsection as a standalone scheduling to ol and in

combination with the enumeration routine chain of the previous subsection to set up chaining

set partitioning problems with up to columns

Vehicle Scheduling at Telebus

Computational Results

We rep ort in this section on computational exp eriences with our vehicle scheduling system

The cluster and tour generation mo dules and the heuristics of Sections and and

the branchandcut solver BC that is describ ed in Chapter of this thesis Our aim is to

investigate two complexes of questions

i Performance What is the p erformance of our vehicle scheduling system on Telebus

instances Can we solve the clustering and chaining set partitioning instances

ii Vehicle Scheduling Do es our system result in a b etter vehicle scheduling Do es cluster

velling distance ITD Do es ing reduce the internal travelling time ITTinternal tra

the chaining set partitioning mo del yield b etter results than the heuristics

Our test set consists of typical Telebus DARPs from the week of April in

stances vv clustering and tt chaining and another for the week of

September instances vv clustering and tt chaining

April September and April September were Saturdays and Sundays resp ec

tively The two weeks dier in the adjustment of feasibility parameters for clusters and

tours Generally sp eaking the April instances represent a restrictive scenario with continued

concatenations limited to a maximum length of only three small detour times etc The

September problems were pro duced in a lib eral setting with more degrees of freedom the

maximum concatenation length was eg doubled to six

Wehave run our vehicle scheduling system on these problems and rep ort in the following three

subsections ab out the results We give statistics on solving the clustering and chaining set

partitioning problems and weinvestigate the relevance of our integer programming approach

for vehicle scheduling at Telebus We do not give detailed statistics for cluster and tour

generation b ecause these steps are not a computational b ottleneck the interested reader can

nd such data in Klostermeier Kuttner

Following the guidelines of Crowder Dembo Mulvey and Jackson Boggs Nash

Powell for rep orting ab out computational exp erimen ts we state that all test runs were

made on a Sun Ultra Sparc Mo del E workstation with MB of main memory running

SunOS that our branchandcut co de BC was written in ANSI C compiled with the Sun

cc compiler and switches fast xO and that we have used the CPLEX Callable

Library V as our LP solver

Our computational results are listed in tables that have the following format Column gives

the name of the problem columns its size in terms of numbers of rows columns and

nonzeros and columns the size after an initial prepro cessing The next two columns give

z rep orts the value of the b est solution that could b e found This number is solution values

aproven optimum when the duality gap is zero which is indicated bya Otherwise we are

left with a nonzero dualitygap z z z where z is the value of the global lower b ound The

following ve columns give statistics on the branchandcut algorithm There are from left

to right the numb er of in and outpivots Pvt cutting planes Cut simplex iterations to

solve the LPs Itn LPs solved LP and the numb er of branchandb ound no des BB The

next ve columns give timings The p ercentage of the total running time sp ent in problem

reduction PP pivoting Pvt separation Cut LPsolution LP and the heuristic Heu

Thelastcolumngives the total running time in CPU seconds

Available at URL httpwwwzibdeborndoerfer

Confer Chapter for an explanation of this concept

Computational Results

Clustering

Table lists our clustering results The rst seven rows of this table corresp ond to the April

instances the next fourteen to the Septemb er instances whichwere solved twice We used a

time limit of seconds to pro duce the results in rows and seconds in rows

We can see from column of the table that the DARP instances that we are considering here

involve or more requests during the week Tuesday is usually a p eak and signicantly

less requests on weekends These numbers were typical for Telebus in The requests were

clustered in all p ossible ways and this resulted in the number of clusters that is rep orted in

column The restrictive parameter settings for April lead to a rather small numb er of feasible

clusters only ab out four times the numb er of requests v is an exceptional instance that

contains extraordinary large collective requests More planning freedom in Septemb er lead

to a fold increase in the numb er of feasible clusters Wenotethattheaverage April cluster

contains three requests while the numb er for Septemb er is four

As the numb er of feasible clusters for April is very small one would exp ect that clusters do

not overlap much and that there are often not many choices to assign requests to clusters

The statistics on preprocessing in columns of rows show that this is indeed so The

extremely large reduction in the number of rows indicates that in particular many requests

can only be assigned in a single way to a cluster either to a single p ossible cluster or in

exactly the same way as some other request The results for September are dierent see

rows We observe also signicant and encouraging problem reductions but not to the

same extreme extent as for the April problems

The trends that we observed in the prepro cessing step contin ue in the branchandcut phase

Largely orthogonal columns and few rows in the April instances translate into simple LPs with

more or less integral solutions The problems could b e solved with a few LPs cutting planes

and branchandb ound no des two even at the ro ot of the searchtree see columns in

rows Iterated prepro cessing played a ma jor role in these computations as can be seen

from the large number of pivots in column Pvt this is a measure for successful prepro cess

ing see Subsection note that the co de sp ent ab out half of the total running time in

problem reduction sums of Timing columns PP and Pvt All in all ab out three minutes of

CPU time were always sucient to solve the easy April problems to proven optimality The

situation is dierent for the Septemb er data The problems are larger and substantial overlap

in the clusters results in highly fractional LPs Signicant computational eort and extensive

branching is required to solve the September problems see columns of rows in

fact three instances could not b e solved completely within seconds But the remaining

duality gaps are so small that any practitioner at the BZA is p erfectly happy with the so

lutions And these results can even be obtained much faster Setting the time limit to only

seconds yields already solutions of very go o d quality see column Gap in rows

The ob jective that we used in the April and Septemb er clustering set partitioning problems

was a mixture of ITD and a p enalty that discourages the clustering of taxi requests servicing

all but the most clusterable taxi requests with individual taxi rides was BZA p olicy at

number of requests and the number of that time Figure compares on its left side the

clusters that were obtained by optimizing this mixed criterion for the September data Note

that the number of taxi clusters that contain only taxi requests is largely identical to the

original numb er of taxi requests ie the taxi requests were essentially left unclustered The

observed reductions are thus solely due to the clustering of bus requests The right side gives

an impression of the reduction of ITD that can b e achieved with a clustering of this typ e

Vehicle Scheduling at Telebus v v v v v v v v v v v v v v v v v v v v v Name a b iei P eod naSnUtaSacM e E del Mo Sparc Ultra Sun a on T seconds CPU in Time tltm xed ielmtbeas h atL ssolv is LP last the ecause b limit time exceeds time otal Ro ws rgnlProblem Original Cols NNEs Ro ws rpocessed Prepro T Cols be able NNEs ovn lseigStP Set Clustering Solving dbeoestopping efore b ed Solutions z Gap Pvt riinn Problems artitioning Branc Cut handCut Itn LP BB PP Pvt Timings Cut LP Heu T Time otal a b b b b b b b b

Computational Results

b er Septemb er Septem Verknüpfungsoptimierung (16. bis 22.9.1996) 15.000 km 1.500 10.000 km 1.000 5.000 km 500 0 km

Tu We Th Fr Sa Su Mo

Mo Di Mi Do Fr Sa So

Mo Tu We Th Fr Sa Su

Requests

Bus Taxi

Bus Durch das Verknüpfen von Fahraufträgen zu Bestellungen

Requests

ITD

Clusters

Bus Bus Taxi

Clusters lassen sich die Besetztkilometer um 20% reduzieren.

Reducing Internal Travelling Distance by Clustering

Figure

Chaining

Wehave used the b est clusterings that we computed in the tests of the previous subsection to

set up two sets of chaining problems Table lists our results for these problems Rows

corresp ond to the April instances rows are for the September chaining problems

The instances for April contain redundant data namely identical rows for every request in

a cluster ie tours are stored by requests not by clusters thus these problems have the

same number of rows as their clustering relatives This is not so for the Septemb er instances

which are stored by clusters In addition wehave also already removed from these instances

all clusters that corresp ond to individual taxi rides These clusters have to b e serviced exactly

in this way with an individual taxi ride and would give rise to row singletons The number

of rows in the Septemb er instances is thus exactly the sum of the heigh ts of the columns for

bus and bus clusters in Figure

The picture for tour optimization has the same avour as the clustering A small number

of tours was pro duced for April more p otential is present in the September data where the

average tour services between four and ve clusters Thinking ab out the p ossible success of

preprocessing one would guess that tours which extend over a long p erio d of time and a large

area of service have a signicantly larger overlap than clusters whichhavealocalcharacter

in space and time Hence it is potentially much more dicult to nd out ab out p ossible

reductions The real situation is even worse Mostly only duplicate tours are eliminated in

the prepro cessing step The chaining problems contain these duplicates in large numbers

b ecause our tour generation pro cedures tend to pro duce unfortunately the same lo cally

promising tours many times The large reduction in the number of rows for the April

instances is solely due to the removal of the duplicates that represent each cluster several

times and to the detection of row singletons that corresp ond to individual taxi rides These

redundancies were already eliminated during the generation of the Septemb er problems and

not a single further rowcouldberemoved there

Small reductions in prepro cessing are a go o d indicator for the computational hardness of a set

partitioning problem and the chaining instances turn out to b e very hard indeed Although

the problems are at best medium scale we can solve none of them to proven optimality

with our branchandcut co de and the duality gaps are disapp ointingly large in comparison

Vehicle Scheduling at Telebus Name t t t t t t t t t t t t t t a b iei P eod naSnUtaSacM e E del Mo Sparc Ultra Sun a on T seconds CPU in Time tltm xed ielmtbeas h atL ssolv is LP last the ecause b limit time exceeds time otal Ro rgnlProblem Original ws Cols NNEs Ro ws rpocessed Prepro Cols T be able NNEs ovn hiigStP Set Chaining Solving dbeoestopping efore b ed Solutions z Gap Pvt riinn Problems artitioning Branc Cut handCut Itn LP BB PP Pvt Timings Cut LP Heu T Time otal a b b b b b b b b b b b b b b

Computational Results

to results for similar applications most notably airline crew scheduling For the September

problems we do not even get close to optimality Lo oking at the pivoting column Pvt we see

that substantial reductions are achieved in the tree search but this do es obviously not suce

In fact the LPs do not only have completely fractional solutions they are also dicult to

solve The average number of simplex pivots is well ab ove and what one can not see

from the table the basis factorizations ll up more and more the longer the algorithm runs

Another problem is the primal LP plunging heuristic that do es not work well An integral

solution has ab out variables at value one for tours in a schedule and if the LP solution

is not strongly biased to an integral one or decomp oses after a few decisions as in clustering

it is not a go o d idea to searchforsuch a solution by iteratively xing v ariables

As these results are as they are wemust unfortunately sp eculate nowwhy this is so We see

three points i Our current column generation pro cess pro duces a set of vehicle schedules

plus some variations of tours using greedy criteria This works well as a heuristic but it

do es not result in a large combinatorial variety of tours and there is no reason to b elieve that

such tours can b e combined in manyways to complete schedules Rather the contrary seems

to be the case One can observe that the heuristics in BC nearly always fail in the chaining

problems If the set partitioning problems that we pro duce have by construction only few

feasible solutions it is not surprising that a branchandcut algorithm gets in to trouble We

remark that one can not comp ensate this aw with simple minded tricks like eg adding

tripp er tours unit columns b ecause these invariably lead to schedules of unacceptable costs

ii There are some reasons whyTelebus DARPs might result in set partitioning problems that

are dicult p er se In comparison to the Homan Padb erg airline test set where our

algorithm BC works well see Chapter the Telebus chaining SPPs have more rows and the

solutions haveamuch larger supp ort iii And mayb e there is a structural dierence b etween

airline crew and bus scheduling Marsten Shepardson also rep ort the computational

hardness of set partitioning problems from bus driver scheduling applications in Helsinki

Vehicle Scheduling

We can not solve the chaining SPPs of Telebus DARPs to optimality but the approximate

solutions that we can obtain are still valuable for vehicle scheduling

Table on the following page gives a comparison of dierent vehicle scheduling metho ds

for the September DARPs Column lists the name of the instance column the day of the

week and column the number of requests The next three columns show the results of a

heuristic vehicle scheduling that used the cluster and tour generators of Sections and

as standalone optimization mo dules There are from left to right the number of clusters

in a heuristic clustering its ITD and the costs of a heuristic vehicle schedule computed

from this clustering We compare these numbers with the results of two set partitioning

for the moment we see in columns and the clustering approaches Skipping column

results of Figure Using this optimized clustering as input for the chaining heuristics

results in vehicle schedules with costs that are rep orted in column The nal column

lists the costs of the vehicle schedules that we computed in Subsection

These results indicate substantial potential savings In our tests the set partitioning clustering

yields less clusters than a heuristic clustering and ab out the same improvementinITD

Heuristic vehicle scheduling based on such a clustering can save DM of op eration costs

per day in comparison to the purely heuristic approach Set partitioning based chaining can

reduce costs by another DM p er day

Vehicle Scheduling at Telebus

Heuristics Set Partitioning

Name Day Requests Clusters Tours Clusters Tours

ITDkm DM DM ITDkm DM

Scheduling anomaly due to heuristic chaining

Table Comparing Vehicle Schedules

Persp ectives

Telebus is an example that mathematical programming techniques can make a signicant

contribution to the solution of largescale transportation problems of the real world We

mention here three further p ersp ectives and refer the reader to Borndorfer Grotschel Lob el

and the references therein for a broader treatmen t of optimization and transp ortation

Telebus Wehave p ointed out in Section that mathematical vehicle scheduling metho ds

as one factor have translated into cost reductions and improvements in service at Telebus

t present the BZA utilizes And the optimization p otential at Telebus is not yet depleted A

only the heuristic mo dules of our scheduling system We have seen in Subsection that

integer programming allows for further cost reductions that have to b e put into practice

Computer Aided Scheduling Automatic scheduling paves the way for a systematic sce

nario analysis not only at the BZA The scheduler of the future will use software planning

to ols based on advanced mathematical metho ds to simulate analyze and anticipate the impli

cations of changing op eration conditions and variations in contractual obligations Computer

aided design CAD has replaced the drawing b oard computer aided manufacturing CAM

controls the factories computer aidedscheduling CAS for logistic systems is just another

step in this direction

Paratransit Berlins Telebus system of to day is only a remainder of a comprehensive

paratransit concept that was develop ed as a part of the seventies eorts to revitalize the

public transp ortation sector The idea to reduce costs and simultaneously improve and extend

service in times and areas of low trac with demand responsive systems was convincing

and immediately tested in a number of pilot schemes in Germany in Friedrichshafen in

Wunstorf near Hannover and with a slightly dierent scop e in Berlin see Figure for

the dimensions that were initially pro jected for Telebus But some years later most of these

systems had either disapp eared or turned into sp ecial purp ose systems And there can b e no

doubt that for instance the handicapp ed only used a system with advance callin p erio ds of

initially three days b ecause there was no other choice The main reason for the lack of success

e been scheduling problems After initial enthusiasm in of dialaride systems seems to hav

every single one of these pro jects the systems were virtually killed by their own success

beyond a critical size

Persp ectives

Figure From a Telebus Pro ject Flyer

But right now the situation is changing and old reasons have kindled new interest in para

transit see eg Sudmersen German for some examples of ongoing pro jects Why

The driving force b ehind renewed p opularity of demand resp onsive systems and many other

developments is the up coming deregularization of the Europ ean public transp ortation sector

according to Article of the Maastricht II treaty of the Europ ean Union see Meyer

German for some background information and a survey of the current situation of public

transp ort in the EU This lawgives a new chance to dialaride typ e systems The future will

show if mathematical programming techniques can help to takeit

Vehicle Scheduling at Telebus

Bibliography of Part

Assad Ball Bo din Golden Routing and scheduling of vehicles and crews Computers

Operations Research

Ball Magnanti Monma Nemhauser Eds Network Routingvolume of Handbooks

in Operations Research and Management Science Elsevier Sci BV Amsterdam

Barnhart Johnson Nemhauser Savelsb ergh Vance BranchandPrice Column

Generation for Solving Huge Integer Programs In Birge Murty pp

Birge Murty Eds Mathematical Programming State of the Art Univ of

Michigan

Bo din Golden Classication in Vehicle Routing and Scheduling Networks

Borndorfer Grotschel Herzog Klostermeier Konsek Kuttner Kurzen mu nicht

Kahlschlag heien Das Beispiel TelebusBehindertenfahrdienst Berlin Preprint SC

KonradZuseZentrum Berlin To app ear in VOP

Borndorfer Grotschel Klostermeier Kuttner a Berliner Telebus bietet Mobilitat fur

Behinderte Der Nahverkehr ZIB Preprint SC

b Optimierung des Berliner Behindertenfahrdienstes DMV Mitteilungen

ZIB PreprintSC

Borndorfer Grotschel Lob el Optimization of Transp ortation Systems PreprintSC

KonradZuseZentrum Berlin To app ear in ACTAFORUM ENGELBERG

CPLEX Using the CPLEX Cal lable Library Suite Taho e Blvd Bldg

Incline Village NV USA CPLEX Optimization Inc

Crowder Demb o Mulvey On Rep orting Computational Exp eriments with Mathe

matical Software ACM Transactions on Math Software

Cullen Jarvis Ratli Set Partitioning Based Heuristics for Interactive Routing

Networks

Daduna Wren Eds ComputerAided Transit Sche duling Lecture Notes in Eco

nomics and Mathematical Systems Springer Verlag

DellAmico Maoli Martello Eds Annotated Bibliographies in Combinatorial

Optimization John Wiley Sons Ltd Chichester

Desrosiers Dumas Solomon Soumis Time Constrained Routing and Scheduling

In Ball Magnanti Monma Nemhauser chapter pp

Avail at URL httpwwwzibdeZIBbibPublic atio ns

Inf avail at URL httpwwwcplexcom

BIBLIOGRAPHY OF PART

Desrosiers Dumas Soumis The Multiple Vehicle DIALARIDE Problem In

Daduna Wren

Hamer Fully Automated Paratransit Scheduling for Large MultiContractor Op era

tions In Preprints of the th Int Workshop on CompAided Sched of Public Transp

Homan Padb erg Solving Airline CrewScheduling Problems by BranchAndCut

Mgmt Sci

Ioachim Desrosiers Dumas Solomon A Request Clustering Algorithm in Do orto

Do or Transp ortation Tech Rep G Ecole des Hautes Etudes Commerciales de

Montreal Cahiers du GERAD

Jackson Boggs Nash Powell Guidelines for Rep orting Results of Computational

Exp eriments Rep ort of the Ad Ho c Committee Math Prog

Klostermeier Kuttner Kostengunstige Disp osition von Telebussen Masters thesis

Tech Univ Berlin

Lap orte Vehicle Routing In DellAmico Maoli Martello chapter pp

Lob el Optimal Vehicle Scheduling in Public Transit PhD thesis Tech Univ Berlin

Marsten Shepardson Exact Solution of Crew Scheduling Problems Using the Set

Partitioning Mo del Recent Successful Applications Networks

Meyer Regionalisierung und Wettb ewerb Der Nahverkehr

Sol Column Generation Techniques for Pickup and Delivery Problems PhD thesis

Tech Univ Eindhoven

Soumis Decomp osition and Column Generation In DellAmico Maoli Martello

chapter pp

Sudmersen Auf Anruf SammelTaxi Der Nahverkehr

Tesch Disposition von AnrufSammeltaxis Deutscher Univ Verlag Wiesbaden

Index of Part

continued concatenation Numb ers

bus at Telebus

feasible time window

bus at Telebus

insertion

internal travelling distance

internal travelling time

accessibility in public transp ortation

load of a vehicle

advance callin p erio d at Telebus

service time

ambulatory at Telebus

start time window

April test set of Telebus DARPs

cluster cluster generation pro cedure

assignment heuristic

cluster decomp osition

for vehicle scheduling

of a vehicle schedule

assistance at Telebus

cluster generation

auxiliary ob jectives at Telebus

by depth rst search

availabilitysee vehicle availability

cluster procedure

feasible state

infeasible predicate

Berliner Senatsverwaltung

multilab el shortest path algorithm

furSoziales

output procedure

fur Wissenschaft Forschung Kult

state elimination

Berliner Zentralausschu

state transition

fur Soziale Aufgab en eV

tail function

break see driver break

terminal predicate

break arc in a transition network

terminal state

break no de in a transition network

time window of a state

bus plot at Telebus

transition network

BZA heuristic

clusterrst schedulesecond algorithm

for vehicle scheduling

clustering

computational results

problem

capacity constraintatTelebus

set partitioning problem

capacityofavehicle

collection cluster

chain tour generation pro cedure

column generatn for vehicle scheduling

chaining

computational results

for chaining

problem

forclustering

set partitioning problem

for vehicle scheduling

cluster

computer aided scheduling

at T elebus

inpublictransportation collection

concatenation cluster concatenation

INDEX OF PART

constraints at Telebus feasible vehicle schedule

availability feasible vehicletour

capacity

folding wheelchair at Telebus

depot

nostop

greedy heuristic

pairing

for vehicle scheduling

precedence

shift

time window

handicapp ed p eoples transp ort

visiting

heuristic vehicle scheduling

continued concatenation cluster

hybrid heuristic

cost reductions at Telebus

for vehicle scheduling

customer at Telebus

customer satisfaction at Telebus

infeasible predicate

inclustergeneration

decomp osition

intourgeneration

approachtovehicle sched

insertion cluster

of a vehicle schedule into clusters

internal travelling distance

demand resp onsive system

internal travellingtime

depot see vehicle dep ot

Intranetz Gesellschaft

dep ot constraintatTelebus

fur Informationslogistik mbH

depth rst search

forclustergeneration

for tour generation

k b est neighb ors heuristic

deregularization in public transp ort

for vehicle scheduling

desired pickup time of a request

KonradZuseZentrum Berlin

dialarideproblem

dialaridesystem

disabledpeople

Lagrangean pricing

do ortodo or transp ortation at Telebus

load of a vehicle

driver break at Telebus

dropo event at Telebus

dropo lo cation of a request

Maastricht I I treaty of theEU

dropo time window of a request

manual planning metho d

for vehicle scheduling

minimal cluster decomp osition see cluster

entitled user at Telebus

decomp osition

event

mobility disabled p eople

dropo

multi commo dityow relaxation

pickup

of a vehicle scheduling problem

event arc in a transition network

multilab el shortest path algorithm

event no de in a transition network

multiple dep ot vehicle scheduling

feasible state

no stop constraintatTelebus

in cluster generation

in tour generation nonfolding wheelchair at Telebus

INDEX OF PART

scheduling see vehicle scheduling O

Senate of Berlin ob jectives at Telebus

auxiliary ob jectives

Dept for Sci Research Culture

customer satisfaction

Dept for So cial Aairs

op eration costs

Senatsverwaltungsee Berliner

op eration costs at Telebus

Senatsverwaltung

output pro cedure

Septemb er test set of Telebus DARPs

in cluster generation

service improvements at Telebus

in tour generation

service time

for a cluster

fora request

pairing constraintatTelebus

set partitioning

paratransit

chaining problem

p eak of demand at Telebus

clustering problem

pickup eventatTelebus

for vehicle scheduling

pickup lo cation of a request

shift constraintatTelebus

pickup time window of a request

shift of op eration at Telebus

piece of work at Telebus

sp ontaneous request at Telebus

precedence constraintatTelebus

staircase assistance at Telebus

public transp ortation

start time window fora cluster

accessibility

state

computer aided scheduling

forclustergeneration

demand resp onsive systems

for tour generation

deregularization

state elimination

for mobility disabled p eople

inclustergeneration

paratransit

intourgeneration

perspectives for the future

state transition

inclustergeneration

intourgeneration

reorganization of Telebus

subscrib er dialaride

request

routing and scheduling problem

at Telebus

desired pickup time

dropo event

tail function

dropo lo cation

inclustergeneration

dropo time window

intourgeneration

pickup event

taxi account system at Telebus

pickuplocation

taxi voucher system at Telebus

pickup time window

Telebus

servicetime

bus

sp ontaneous

bus

reunication of Germany

advance callin p erio d

eects on Telebus

ambulatory

road network of Berlin

assistance

availability constraint

break see driver break

saving p otentials in vehicle sc heduling

schedule see vehicle schedule busplot

INDEX OF PART

pickup event capacity constraint

pickup lo cation capacity of a vehicle

pickup time window cluster

piece of work collection

precedence constraint concatenation

project continued concatenation

cost reductions feasible time window

reorganization of the center insertion

results internal travelling distance

service improvements internal travelling time

reorganization of the center load of a vehicle

request servicetime

desired pickuptime start time window

dropo event computer system

dropo lo cation constraints

dropo time window availability

pickup event capacity

pickuplocation depot

pickup time window nostop

service time pairing

sp ontaneous precedence

road network shift

schedulesee vehicle schedule time window

scheduling see vehicle scheduling visiting

service improvements cost reductions

service time of a request customer

shift constraints customer satisfaction

shift of op eration dep ot constraint

sp ontaneousrequest desired pickup time

staircase assistance dialaride problem

taxi account system dialaride system

taxi voucher system do ortodo or transp ortation

Teletaxi driverbreak

time dropo event

time window constraint dropo lo cation

tour see vehicle tour dropo time window

transp ortation request see request eects of reunication of Germany

vehicle entitleduser

vehicle capacity folding wheelchair

vehicle dep ot no stop constraint

nonfolding wheelchair vehicle provider

ob jectives vehicle renting

auxiliary ob jectives vehicle schedule

customer satisfaction vehicle scheduling

operationcosts vehicle scheduling problem

pairing constraint Apriltest set

peakof demand Septembertest set

INDEX OF PART

tour end arc vehicletour

tour end no de vehicle types

bus

tour start arc

bus

tour start no de

Teletaxi

transp ortation request see request see

visiting constraint

request

Teletaxi at Telebus

terminal predicate

vehicle at Telebus

in cluster generation

vehicle availability constraint

in tour generation

at Telebus

terminal state

vehicle dep ot at Telebus

in cluster generation

vehicle provider at Telebus

in tour generation

vehicle renting at Telebus

time sweep greedy heuristic

vehicle schedule at Telebus

for vehicle scheduling

vehicle scheduling

time window constraintatTelebus

algorithm

time window of a state

assignment heuristic

in cluster generation

at Telebus

in tour generation

BZAheuristic

tour see vehicle tour see vehicle tour

chaining

tour end arc in a transition network

computational results

tour end no de in a transition network

problem

tour generation

set partitioningproblem

by depth rst search

clustergeneration

chain tour generation pro cedure

by depth rst search

feasiblestate

cluster pro cedure

infeasible predicate

feasiblestate

output pro cedure

infeasible predicate

state elimination

multilab el shortest path alg

state transition

output procedure

tail function

state elimination

terminal predicate

state transition

terminal state

tail function

time window ofa state

terminal predicate

transition network

terminalstate

waiting

time window of a state

tour start arc in a transition network

clustering

tour start no de in a transition network

computational results

tourbytour greedy heuristic

problem

for vehicle scheduling

set partitioning problem

transition network

column generation

breakarc

computational results

breaknode

Apriltest set

event arc

for chaining

event node

for clustering

forclustergeneration

for tour generation Septembertest set

INDEX OF PART

at Telebus decomp osition approach

Teletaxi feasible vehicle schedule

visiting constraintatTelebus feasible vehicle tour

heuristic

assignment

waitingintourgeneration

BZA

WheelTrans Service in Toronto

hybrid

k b est neighbors

time sweep greedy

zillion

tourbytour greedy

hybrid heuristic

k b est neighborsheuristic

manual planning metho d

problem

at Telebus

multi commo dityow relaxation

saving p otentials

set partitioning approach

time sweep greedy heuristic

tour generation

by depth rst search

chain tour generation pro cedure

feasible state

infeasible predicate

output procedure

state elimination

state transition

tail function

terminal predicate

terminal state

time window ofa state

transition network

waiting

tourbytour greedy heuristic

transition network

breakarc

breaknode

event arc

event node

tourendarc

tourendnode

tour start arc

tour start no de

vehicle tour at Telebus

vehicle typ es

bus bus

INDEX OF PART

Index of Symbols

Symb ols C

a bt a t b t C k graphincidence matrix

powerset of the set S of an o dd hole

C n k antiweb ina graph

Numb ers

c v capacityforambulatories

vector ofallones

c v capacit y for folding wheelchairs

c v capacity

for nonfolding wheelchairs

Ak edgeno de incidence matrix

c v total capacity

of anoddhole

c v capacity for wheelchairs

AG edgeno de incidence matrix

CAS computer aided scheduling

of a graph G

C n k web in a graph

abl A antiblo cker of a matrix A

CHAIN Telebus chaining problem

abl P antiblo cker of a p olytop e P

H chromatic number

abl A integral part of abl A

of a hyp ergraph

A algebra

Gweighted clique covering number

acs airline crew scheduling problems

of a graph G

a C vehicle load at cluster no de i

Gweighted coloring number

A minor of a matrix A

of a graph G

G stabilitynumber

C comp osition of circulants in a graph

of a graph G

CLUSTER Telebus clustering problem

Gw eighted stabilitynumber

cone S nonnegativehullofasetS

of a graph G

conv S convex hull of a set S ix

ASP acyclic sub digraph problem

core A core of a matrix A

AW n t q generalized antiweb

CPP clique partitioning problem

union of disjoint sets

B GA bipartite auxiliary graph

in the GLS algorithm

DARP Telebus dialaride problem

BC set partitioning solver

symmetric dierence op erator

BCP bipartite no de covering problem

C C multicut in a graph

A rank of the matrix A

W cut ingraph

Bi binomial distribution

diam G diameter of a graph G

m iid trials probability for

D complete digraph on n nodes

bl A blo cker of a matrix A

bl P blo cker of a p olytop e P

BMP bipartite matching problem E matrix of all ones

BZA Berliner Zentralausschu e unit vector

fur Soziale Aufgab en eV EU Europ ean Union

INDEX OF SYMBOLS

F K

F set of fractional variables

k MCP k multicutproblem

in an LP solution

K complete bipartite graph

FCP fractional covering problem

m no des left n no des right

f vector of the form e e

K complete graph on n nodes

i i

FPP fractional packing problem

LG line graph of a graph G

G group of pieces of work

maximum numb er of nonzeros

GA column intersection graph

in a column of a matrix

asso ciated to a matrix A

LOP linear ordering problem

GA fractional intersection graph

G ij graph G with edge ij removed

G j graph G with no de j contracted

M G semidenite fractional

G G intersection of two graphs

set packing matrix cone

G G union of two graphs

M G fractional

v set of endno des of arcs that go out

set packing matrix cone

from v

M G semidenite fractional

Geo geometric distribution

set packing matrix cone

with parameter

MCP max cut problem

G conict graph

M Gsetpacking matrixcone

GA aggregated fractional intersection

MKP multiple knapsack problem

graph

average numb er of nonzeros

GG conict graph

in a column of a matrix

of a rank set packing relaxation

H Ahyp ergraph

N natural numbers ix

asso ciated to a matrix A

N G pro jected fractional

H G fractional set packing cone

set packing matrix cone

H G set packing cone

N natural numb ers without ix

N G pro jected semidenite fractional

set packing matrix cone

I identity matrix

N index

IA indep endence system

of a graph

asso ciated to a matrix A

of an inequality

I v set of items

of matrix inequality

in itemknapsack conguration v

N index

ij edge or arc in a graph or digraph

of a graph

i j edge or arc in a graph or digraph

of an inequality

I n n identity matrix

of matrix inequality

IP integer program

N G k th pro jected fractional set packing

ITD internal travelling distance

matrixcone

ITT internal travellingtime

N G k th pro jected semidenite fractional

set packing matrix cone

Gmaximum size

J incidence matrix of a degenerate pro jec

tive plane with n points of a matching in a graph G

INDEX OF SYMBOLS

O Q

Q rational numbersix

Gweighted clique number

QA fractional covering p olyhedron

of a graph G

QA fractional set covering p olyhedron

Q nonnegative rational numbers ix

Q Asetcovering p olytop e

P probability distribution

QSP quadratic fractional

P A fractional packing p olytop e

set packing problem

P A fractional set packing p olytop e

QSP semidenite relaxation

P A b p olytop e

of the set packing problem

asso ciated to the system Ax b

QSP semidenite fractional

pv pickup lo cation

set packing problem

pv dropolocation

P empty columnreduction

R real numbersix

P emptyrow reduction

R nonnegativerealnumbersix

P row singleton reduction

density ofa matrix

P dominated column reduction

RSF recursive smallest rst semiheuristic

P duplicate column reduction

P dominated row reduction

P duplicate row reduction

P row clique reduction

S set of nite sequences from set S

P parallel column reduction

S p olar of a set S

P symmetric dierence reduction

SCP set covering problem

P symmetric dierence reduction

SenSoz

P column singleton reduction

Berliner Senatsverwaltung

P reduced cost xing

furSoziales

P probing

Senate of Berlins

P A fractional

Dept for So cial Aairs

SenWiFoKult

set partitioning p olytop e

Berliner Senatsverwaltung

P acyclic sub digraph p olytop e

ASP

ur Wissenschaft Forschung Kult f

P clique partitioning p olytop e

CPP

prop erty of a matrix

Senate of Berlins

P set packingpolytope

Dept for Sci Research Culture

prop erty of a matrix

SPP set partitioning problem

P A set packing p olytop e

SSP

P G set packing p olytop e

set packing problem

A set partitioning p olytop e P

stable set problem

P polytope

asso ciated to an integer program

P indep endence system p olytop e

ISP

T C cluster start time window

P k multicut p olytop e

k MCP

T v dropo time window

P linear ordering p olytop e

LOP

T v pickup time window

P max cut p olytop e

t v desired pickup time

MCP

P multiple knapsack p olytop e

t C service time for a cluster

MKP

P antidominant

pickup service time t v

SSP

t v dropo service time of a set packing p olytop e i

INDEX OF SYMBOLS

t Telebus DARP W w group of pieces of work

April chaining instance W wheel

t Telebus DARP w oset in a set partitioning problem

w reducedcosts of anLPsolution September chaining instance

tC latest cluster start time

tGmaximum duration of a piece of work

x optimal solution of an LP

of group G

X random variable counting sequence

tv latest pickuptime

intersections

tv latest dropo time

tC earliest cluster start time

t v earliest pickup time

Y random variable counting op erations in

tv earliest dropo time

lexicographic comparisons

TDI total dual integrality

THG semidenite relaxation

of the set packing p olytop e asso ciated

Z integer numbersix

to a graph G

Z nonnegativeinteger numbersix

T C feasible time interval

z optimal ob jectivevalue of an LP

at cluster no de i

ZIB KonradZuseZentrum Berlin

t vehicle travelling time

U homogenized unit cub e

V set of break no des

V set of pickup event nodes

V set of dropo event no des

v Telebus DARP

April clustering instance

v Telebus DARP

Septemb er clustering instance

vert P vertices of a p olyhedron P

V set of tour start no des

V setof tourendnodes

v pickupnode

v droponode

V G set of subgraphs of a graph G

with at most k nodes

v breaknode

v tourstart node

v tourendnode

v vehicle typ e of a piece of work

W set of all available pieces of work

W n t q generalized antiweb

set of feasible pieces of work W v

W v set of feasible pieces of work i

INDEX OF SYMBOLS

Curriculum Vitae

Ralf Borndorfer

geb oren am August in Munster

Grundschule in Oerlinghausen

HansEhrenb ergGymnasium in BielefeldSennestadt

Grundwehrdienst b ei der Panzerbrigade in Augustdorf

Studium der Wirtschaftsmathematik an der Universitat Augsburg

Dez Diplom in Wirtschaftsmathematik mit Schwerpunkt Optimierung

Betreuer Prof Dr Martin Grotschel

Wissenschaftlicher Mitarb eiter an der Technischen Universitat Berlin

Betreuer Prof Dr Martin Grotschel

seit

April Wissenschaftlicher Mitarb eiter am KonradZuseZentrum Berlin

Curriculum Vitae

Ralf Borndorfer

b orn on August in Munster

Elementary Scho ol in Oerlinghausen

HansEhrenb ergGymnasium in BielefeldSennestadt

Military Service at the Armored Brigade in Augustdorf

Studies of Mathematics with Economics at the University of Augsburg

Dec Masters Degree in Mathematics with Economics

Sup ervisor Prof Dr Martin Grotschel

Teaching Assistant at the Technical University of Berlin

Sup ervisor Prof Dr Martin Grotschel

since

April Scientic Assistantat the KonradZuseZentrum Berlin